## 17Calculus Precalculus - Equations of Lines

### Functions

Functions

Polynomials

Rational Functions

Matrices

Systems

Trigonometry

Complex Numbers

### Practice

Calculus 1 Practice

Calculus 2 Practice

Practice Exams

### Articles

Line Equations

slope

$$\displaystyle{ m = \frac{y_2 - y_1}{x_2 - x_1} }$$

point-slope form

$$y-y_1 = m(x-x_1)$$

slope-intercept form

$$y=mx+b$$

general form

$$Ax + By = C$$

Before we get down to the details of lines, let's watch this video to remind us of the slope of a line.

### Prof Leonard - Introduction to the Slope of a Line [12min-29secs]

video by Prof Leonard

In order to determine an equation of a line, you need only two pieces of information, the slope and one point on the line. If you are given two points and no slope, use the slope equation $$\displaystyle{ m = \frac{y_2 - y_1}{x_2 - x_1} }$$ to find the slope and then use one of the points (either one, its doesn't matter, you will get the same answer no matter which point you choose).

Once you have the slope, usually called m, and one point, $$(x_1,y_1)$$, you can find the equation of the line using two possible equations.
1. The one that is the easiest and requires the least amount of algebra is the point-slope form, $$y-y_1 = m(x-x_1)$$. Notice that this is just the slope equation in a little different form.

### Prof Leonard - Using the Point-Slope Equation of a Line [24min-6secs]

video by Prof Leonard

2. The second, which works too, is to use the slope-intercept form, $$y=mx+b$$ and solve for $$b$$. This seems to be the preferred method of most students.
For your final answer, most instructors prefer the slope-intercept form $$y=mx+b$$. This is the required form for answers on this site. However, you need to check with your instructor to see what they expect. Here is a video on why we need the slope-intercept form.

### Prof Leonard - Why We Need Slope-Intercept Form [23min-55secs]

video by Prof Leonard

Slope-Intercept Form of a Line

Adjust the sliders for the slope m and the y-intercept b to see the plot and the equation change.

As we said above, your goal is to get the slope and one point on the line in order to find the equation of the line. Start with these problems to get some practice on this.

Instructions - - Unless otherwise instructed, find the equation of the line, in slope-intercept form, with the given slope passing through the given point.

Find the equation of the line, in slope-intercept form, with slope $$-2/3$$ passing through the point $$(-4, 6)$$.

Problem Statement

Find the equation of the line, in slope-intercept form, with slope $$-2/3$$ passing through the point $$(-4, 6)$$.

$$y = -2x/3 +10/3$$

Problem Statement

Find the equation of the line, in slope-intercept form, with slope $$-2/3$$ passing through the point $$(-4, 6)$$.

Solution

### 2710 video

$$y = -2x/3 +10/3$$

Log in to rate this practice problem and to see it's current rating.

Find the equation of the line, in slope-intercept form, with slope $$-2/3$$ passing through the point $$(1, -1)$$.

Problem Statement

Find the equation of the line, in slope-intercept form, with slope $$-2/3$$ passing through the point $$(1, -1)$$.

$$y = -2x/3 - 1/3$$

Problem Statement

Find the equation of the line, in slope-intercept form, with slope $$-2/3$$ passing through the point $$(1, -1)$$.

Solution

 We are given the slope $$m = -2/3$$ and a point on the line $$(1, -1)$$. Let's use the point-slope form of the equation of the line, $$y-y_1 = m(x-x_1)$$ $$y-(-1) = (-2/3)( x- 1)$$ $$y + 1 = -2x/3 + 2/3$$ $$y = -2x/3 + 2/3 - 1$$ Final Answer: $$y = -2x/3 - 1/3$$ Here is how we would use the slope-intercept form to solve this problem. $$y = mx + b \to -1 = (-2/3)(1) + b$$ $$b = -1 + 2/3 = -1/3$$ Final Answer: $$y = -2x/3 - 1/3$$

$$y = -2x/3 - 1/3$$

Log in to rate this practice problem and to see it's current rating.

Find the equation of the line, in slope-intercept form, with slope $$2$$ passing through the point $$(-1, -6)$$.

Problem Statement

Find the equation of the line, in slope-intercept form, with slope $$2$$ passing through the point $$(-1, -6)$$.

$$y = 2x - 4$$

Problem Statement

Find the equation of the line, in slope-intercept form, with slope $$2$$ passing through the point $$(-1, -6)$$.

Solution

### 2711 video

$$y = 2x - 4$$

Log in to rate this practice problem and to see it's current rating.

Find the equation of the line, in slope-intercept form, with slope $$3$$ passing through the point $$(-2, 3)$$.

Problem Statement

Find the equation of the line, in slope-intercept form, with slope $$3$$ passing through the point $$(-2, 3)$$.

$$y = 3x + 9$$

Problem Statement

Find the equation of the line, in slope-intercept form, with slope $$3$$ passing through the point $$(-2, 3)$$.

Solution

 We are given the slope $$m = 3$$ and a point on the line at $$(-2, 3)$$. Let's use the point-slope form $$y-y_1 = m(x - x_1)$$. $$y-3 = 3( x - (-2))$$ $$y - 3 = 3x + 6$$ $$y = 3x + 6 + 3$$ Final Answer: $$y = 3x + 9$$ If we used the slope-intercept form to solve this problem, we would have this. $$y = mx + b \to 3 = 3(-2) + b$$ $$b = 3 + 6 \to b = 9$$ Final Answer: $$y = 3x + 9$$

$$y = 3x + 9$$

Log in to rate this practice problem and to see it's current rating.

Even though your goal is to have the slope and one point on the line, you will not usually be given this exact information in order to find the equation of the line. Sometimes you are given two points. From that information you need to determine the slope and then use one of the points to get the equation of the line. These problems will give you practice with this.

Instructions - - Unless otherwise instructed, find the equation of the line that passes through the two given points. Give your answer in slope-intercept form.

Find the equation of the line in slope-intercept form that passes through $$(-3,7)$$ and $$(5,-1)$$.

Problem Statement

Find the equation of the line in slope-intercept form that passes through $$(-3,7)$$ and $$(5,-1)$$.

$$y=-x+4$$

Problem Statement

Find the equation of the line in slope-intercept form that passes through $$(-3,7)$$ and $$(5,-1)$$.

Solution

### 2712 video

$$y=-x+4$$

Log in to rate this practice problem and to see it's current rating.

Find the equation of the line in slope-intercept form that passes through the points $$(3,-2)$$ and $$(2,-1)$$.

Problem Statement

Find the equation of the line in slope-intercept form that passes through the points $$(3,-2)$$ and $$(2,-1)$$.

$$y=-x+1$$

Problem Statement

Find the equation of the line in slope-intercept form that passes through the points $$(3,-2)$$ and $$(2,-1)$$.

Solution

### 2713 video

$$y=-x+1$$

Log in to rate this practice problem and to see it's current rating.

Find the equation of the line in slope-intercept form that contains $$(5,0)$$ and $$(-4,3)$$.

Problem Statement

Find the equation of the line in slope-intercept form that contains $$(5,0)$$ and $$(-4,3)$$.

$$y = -x/3 + 5/3$$

Problem Statement

Find the equation of the line in slope-intercept form that contains $$(5,0)$$ and $$(-4,3)$$.

Solution

### 2714 video

$$y = -x/3 + 5/3$$

Log in to rate this practice problem and to see it's current rating.

Determine the equation of the line passing through the points $$(1, 1)$$ and $$(5, -1)$$, giving your answer in slope-intercept form.

Problem Statement

Determine the equation of the line passing through the points $$(1, 1)$$ and $$(5, -1)$$, giving your answer in slope-intercept form.

$$y = -x/2 + 3/2$$

Problem Statement

Determine the equation of the line passing through the points $$(1, 1)$$ and $$(5, -1)$$, giving your answer in slope-intercept form.

Solution

### 2715 video

$$y = -x/2 + 3/2$$

Log in to rate this practice problem and to see it's current rating.

Determine the equation of the line passing through the points $$(-1, 1)$$ and $$(1, 7)$$, giving your answer in standard form $$Ax + By = C$$.

Problem Statement

Determine the equation of the line passing through the points $$(-1, 1)$$ and $$(1, 7)$$, giving your answer in standard form $$Ax + By = C$$.

$$3x - y = -4$$

Problem Statement

Determine the equation of the line passing through the points $$(-1, 1)$$ and $$(1, 7)$$, giving your answer in standard form $$Ax + By = C$$.

Solution

### 2716 video

$$3x - y = -4$$

Log in to rate this practice problem and to see it's current rating.

Determine the equation of the line passing through $$(-2, -3)$$ and $$(4, -2)$$. Give your answer in slope-intercept form.

Problem Statement

Determine the equation of the line passing through $$(-2, -3)$$ and $$(4, -2)$$. Give your answer in slope-intercept form.

$$y = x/6 - 8/3$$

Problem Statement

Determine the equation of the line passing through $$(-2, -3)$$ and $$(4, -2)$$. Give your answer in slope-intercept form.

Solution

### 2717 video

$$y = x/6 - 8/3$$

Log in to rate this practice problem and to see it's current rating.

Find the equation of the line that goes through $$(-3, 5)$$ and $$(2, 8)$$. Use the point-slope form $$y - y_1 = m(x - x_1)$$ and the second point. Give your answer in slope-intercept form.

Problem Statement

Find the equation of the line that goes through $$(-3, 5)$$ and $$(2, 8)$$. Use the point-slope form $$y - y_1 = m(x - x_1)$$ and the second point. Give your answer in slope-intercept form.

$$y = 3x/5+ 34/5$$

Problem Statement

Find the equation of the line that goes through $$(-3, 5)$$ and $$(2, 8)$$. Use the point-slope form $$y - y_1 = m(x - x_1)$$ and the second point. Give your answer in slope-intercept form.

Solution

### 2718 video

$$y = 3x/5+ 34/5$$

Log in to rate this practice problem and to see it's current rating.

Determine the equation of the line with an x-intercept of $$3$$ and a y-intercept of $$-4$$. Give your answer in slope-intercept form.

Problem Statement

Determine the equation of the line with an x-intercept of $$3$$ and a y-intercept of $$-4$$. Give your answer in slope-intercept form.

$$y = 4x/3 - 4$$

Problem Statement

Determine the equation of the line with an x-intercept of $$3$$ and a y-intercept of $$-4$$. Give your answer in slope-intercept form.

Solution

### 2719 video

$$y = 4x/3 - 4$$

Log in to rate this practice problem and to see it's current rating.

Okay, before going on, here is great video on what we are going to discuss next, parallel and perpendicular lines.

### Finding Equations of Parallel and Perpendicular Lines

Parallel Lines

When two lines are parallel, their slopes are equal. So, for two lines in slope intercept form $$y = mx + b_1$$ and $$y=mx + b_2$$, the slopes $$m$$ are the same but $$b_1 \neq b_2$$ (unless of course they are the same line). One of the nice things about the slope intercept form, $$y = mx + b$$ is that you can pull the slope $$m$$ and the y-intercept $$b$$ directly from the equation without having to do any work. For the two parallel lines, the slopes are the same but they cross the y-axis at different points.

Instructions - - Unless otherwise instructed, find the equation of the line in slope-intercept parallel to the given line going through the given point.

Find the equation of the line in slope-intercept parallel to $$3x-y/4 = 2$$ going through the point $$(1/6,-8)$$.

Problem Statement

Find the equation of the line in slope-intercept parallel to $$3x-y/4 = 2$$ going through the point $$(1/6,-8)$$.

$$y=12x-10$$

Problem Statement

Find the equation of the line in slope-intercept parallel to $$3x-y/4 = 2$$ going through the point $$(1/6,-8)$$.

Solution

### 2720 video

$$y=12x-10$$

Log in to rate this practice problem and to see it's current rating.

Write the equation of the line that is parallel to $$x - 2y = 6$$ and passes through the point $$(-4, 3)$$, giving your answer in slope-intercept form.

Problem Statement

Write the equation of the line that is parallel to $$x - 2y = 6$$ and passes through the point $$(-4, 3)$$, giving your answer in slope-intercept form.

$$y = x/2 + 5$$

Problem Statement

Write the equation of the line that is parallel to $$x - 2y = 6$$ and passes through the point $$(-4, 3)$$, giving your answer in slope-intercept form.

Solution

### 2721 video

$$y = x/2 + 5$$

Log in to rate this practice problem and to see it's current rating.

Determine the equation of the line that is parallel to $$y = -5x + 4$$ and passes through $$(-1, 4)$$, giving your answer in slope-intercept form.

Problem Statement

Determine the equation of the line that is parallel to $$y = -5x + 4$$ and passes through $$(-1, 4)$$, giving your answer in slope-intercept form.

$$y = -5x - 1$$

Problem Statement

Determine the equation of the line that is parallel to $$y = -5x + 4$$ and passes through $$(-1, 4)$$, giving your answer in slope-intercept form.

Solution

### 2722 video

$$y = -5x - 1$$

Log in to rate this practice problem and to see it's current rating.

Determine the equation of the line that is parallel to $$3x+2y = 12$$ and passes through $$(2, -5)$$, giving your answer in slope-intercept form.

Problem Statement

Determine the equation of the line that is parallel to $$3x+2y = 12$$ and passes through $$(2, -5)$$, giving your answer in slope-intercept form.

$$y = -3x/2 + 6$$

Problem Statement

Determine the equation of the line that is parallel to $$3x+2y = 12$$ and passes through $$(2, -5)$$, giving your answer in slope-intercept form.

Solution

### 2723 video

$$y = -3x/2 + 6$$

Log in to rate this practice problem and to see it's current rating.

Find the equation of the line parallel to the line $$y = 2x + 1$$ that passes through the point $$(1,5)$$, giving your answer in slope-intercept form.

Problem Statement

Find the equation of the line parallel to the line $$y = 2x + 1$$ that passes through the point $$(1,5)$$, giving your answer in slope-intercept form.

$$y = 2x + 3$$

Problem Statement

Find the equation of the line parallel to the line $$y = 2x + 1$$ that passes through the point $$(1,5)$$, giving your answer in slope-intercept form.

Solution

### 2724 video

$$y = 2x + 3$$

Log in to rate this practice problem and to see it's current rating.

Find the equation of the line parallel to the line $$y = 3x - 4$$ that passes through the point $$(6, 4)$$, giving your answer in slope-intercept form.

Problem Statement

Find the equation of the line parallel to the line $$y = 3x - 4$$ that passes through the point $$(6, 4)$$, giving your answer in slope-intercept form.

$$y = 3x - 14$$

Problem Statement

Find the equation of the line parallel to the line $$y = 3x - 4$$ that passes through the point $$(6, 4)$$, giving your answer in slope-intercept form.

Solution

### 2725 video

$$y = 3x - 14$$

Log in to rate this practice problem and to see it's current rating.

Find the equation of the line parallel to the line $$y = 3x/2 + 3$$ that passes through the point $$(4, -3)$$, giving your answer in slope-intercept form.

Problem Statement

Find the equation of the line parallel to the line $$y = 3x/2 + 3$$ that passes through the point $$(4, -3)$$, giving your answer in slope-intercept form.

$$y = 3x/2 - 9$$

Problem Statement

Find the equation of the line parallel to the line $$y = 3x/2 + 3$$ that passes through the point $$(4, -3)$$, giving your answer in slope-intercept form.

Solution

### 2726 video

$$y = 3x/2 - 9$$

Log in to rate this practice problem and to see it's current rating.

Find the equation of the line in slope-intercept parallel to $$x+7y=4$$ through the point $$(7,9)$$.

Problem Statement

Find the equation of the line in slope-intercept parallel to $$x+7y=4$$ through the point $$(7,9)$$.

$$y=-x/7+10$$

Problem Statement

Find the equation of the line in slope-intercept parallel to $$x+7y=4$$ through the point $$(7,9)$$.

Solution

### 2727 video

$$y=-x/7+10$$

Log in to rate this practice problem and to see it's current rating.

Perpendicular Lines

In the case of perpendicular lines, if the equation of one line is $$y = mx + b_1$$, the equation of the other is $$y=-x/m+b_2$$ So the slope of the second line is said to be the negative reciprocal of the first line.

Okay, time for the remaining practice problems.

Instructions - - Unless otherwise instructed, Find the equation of the line, in slope-intercept form, perpendicular to the given line going through the given point.

Find the equation of the line, in slope-intercept form, perpendicular to $$9x+12y=2$$ containing the point $$(15,7)$$.

Problem Statement

Find the equation of the line, in slope-intercept form, perpendicular to $$9x+12y=2$$ containing the point $$(15,7)$$.

$$y=4x/3 - 13$$

Problem Statement

Find the equation of the line, in slope-intercept form, perpendicular to $$9x+12y=2$$ containing the point $$(15,7)$$.

Solution

### 2728 video

$$y=4x/3 - 13$$

Log in to rate this practice problem and to see it's current rating.

Find the equation of the line, in slope-intercept form, perpendicular to $$10x+2y=9$$ going through the point $$(20,3)$$.

Problem Statement

Find the equation of the line, in slope-intercept form, perpendicular to $$10x+2y=9$$ going through the point $$(20,3)$$.

$$y = x/5 - 1$$

Problem Statement

Find the equation of the line, in slope-intercept form, perpendicular to $$10x+2y=9$$ going through the point $$(20,3)$$.

Solution

### 2729 video

$$y = x/5 - 1$$

Log in to rate this practice problem and to see it's current rating.

Determine the equation of the line that is perpendicular to $$x - 4y = 15$$ and passes through $$(3, 1)$$, giving your answer in slope-intercept form.

Problem Statement

Determine the equation of the line that is perpendicular to $$x - 4y = 15$$ and passes through $$(3, 1)$$, giving your answer in slope-intercept form.

$$y = -4x + 13$$

Problem Statement

Determine the equation of the line that is perpendicular to $$x - 4y = 15$$ and passes through $$(3, 1)$$, giving your answer in slope-intercept form.

Solution

### 2730 video

$$y = -4x + 13$$

Log in to rate this practice problem and to see it's current rating.

Write the equation of the line that contains the point $$(3, 5)$$ and is perpendicular to the line $$y = -3x + 2$$, giving your answer in slope-intercept form.

Problem Statement

Write the equation of the line that contains the point $$(3, 5)$$ and is perpendicular to the line $$y = -3x + 2$$, giving your answer in slope-intercept form.

$$y = x/3 + 4$$

Problem Statement

Write the equation of the line that contains the point $$(3, 5)$$ and is perpendicular to the line $$y = -3x + 2$$, giving your answer in slope-intercept form.

Solution

### 2731 video

$$y = x/3 + 4$$

Log in to rate this practice problem and to see it's current rating.

Determine the equation of the line that is perpendicular to $$y = 2x + 1$$ and passes through $$(-4, 5)$$, giving your answer in slope-intercept form.

Problem Statement

Determine the equation of the line that is perpendicular to $$y = 2x + 1$$ and passes through $$(-4, 5)$$, giving your answer in slope-intercept form.

$$y = -x/2 + 3$$

Problem Statement

Determine the equation of the line that is perpendicular to $$y = 2x + 1$$ and passes through $$(-4, 5)$$, giving your answer in slope-intercept form.

Solution

### 2732 video

$$y = -x/2 + 3$$

Log in to rate this practice problem and to see it's current rating.

Find the equation of the line perpendicular to the line $$y = 5x - 3$$ that passes through the point $$(2, -1)$$, giving your answer in slope-intercept form.

Problem Statement

Find the equation of the line perpendicular to the line $$y = 5x - 3$$ that passes through the point $$(2, -1)$$, giving your answer in slope-intercept form.

$$y = -x/5 - 3/5$$

Problem Statement

Find the equation of the line perpendicular to the line $$y = 5x - 3$$ that passes through the point $$(2, -1)$$, giving your answer in slope-intercept form.

Solution

### 2733 video

$$y = -x/5 - 3/5$$

Log in to rate this practice problem and to see it's current rating.

Find the equation of the line perpendicular to the line $$y = 3x/4 - 1$$ that passes through the point $$(8, -3)$$, giving your answer in slope-intercept form.

Problem Statement

Find the equation of the line perpendicular to the line $$y = 3x/4 - 1$$ that passes through the point $$(8, -3)$$, giving your answer in slope-intercept form.

$$y = -4x/3 + 23/3$$

Problem Statement

Find the equation of the line perpendicular to the line $$y = 3x/4 - 1$$ that passes through the point $$(8, -3)$$, giving your answer in slope-intercept form.

Solution

### 2734 video

$$y = -4x/3 + 23/3$$

Log in to rate this practice problem and to see it's current rating.

Find the equation of the line perpendicular to the line $$y = -2x/3 + 4$$ that passes through the point $$(6, -2)$$, giving your answer in slope-intercept form.

Problem Statement

Find the equation of the line perpendicular to the line $$y = -2x/3 + 4$$ that passes through the point $$(6, -2)$$, giving your answer in slope-intercept form.

$$y = 3x/2 - 11$$

Problem Statement

Find the equation of the line perpendicular to the line $$y = -2x/3 + 4$$ that passes through the point $$(6, -2)$$, giving your answer in slope-intercept form.

Solution

### 2735 video

$$y = 3x/2 - 11$$

Log in to rate this practice problem and to see it's current rating.

Find the equation of the line in slope-intercept perpendicular to $$y=2x+1$$ through the point $$(2,0)$$.

Problem Statement

Find the equation of the line in slope-intercept perpendicular to $$y=2x+1$$ through the point $$(2,0)$$.

$$y = -x/2 + 1$$

Problem Statement

Find the equation of the line in slope-intercept perpendicular to $$y=2x+1$$ through the point $$(2,0)$$.

Solution

### 2736 video

$$y = -x/2 + 1$$

Log in to rate this practice problem and to see it's current rating.

Find the equation of the line in slope-intercept perpendicular to $$y=-9x+5$$ through the point $$(3,9)$$.

Problem Statement

Find the equation of the line in slope-intercept perpendicular to $$y=-9x+5$$ through the point $$(3,9)$$.

$$y = x/9 + 26/3$$

Problem Statement

Find the equation of the line in slope-intercept perpendicular to $$y=-9x+5$$ through the point $$(3,9)$$.

Solution

### 2737 video

$$y = x/9 + 26/3$$

Log in to rate this practice problem and to see it's current rating.

### equations of lines 17calculus youtube playlist

Really UNDERSTAND Precalculus

### Trig Formulas

The Unit Circle

The Unit Circle [wikipedia]

Basic Trig Identities

Set 1 - basic identities

$$\displaystyle{ \tan(t) = \frac{\sin(t)}{\cos(t)} }$$

$$\displaystyle{ \cot(t) = \frac{\cos(t)}{\sin(t)} }$$

$$\displaystyle{ \sec(t) = \frac{1}{\cos(t)} }$$

$$\displaystyle{ \csc(t) = \frac{1}{\sin(t)} }$$

Set 2 - squared identities

$$\sin^2t + \cos^2t = 1$$

$$1 + \tan^2t = \sec^2t$$

$$1 + \cot^2t = \csc^2t$$

Set 3 - double-angle formulas

$$\sin(2t) = 2\sin(t)\cos(t)$$

$$\displaystyle{ \cos(2t) = \cos^2(t) - \sin^2(t) }$$

Set 4 - half-angle formulas

$$\displaystyle{ \sin^2(t) = \frac{1-\cos(2t)}{2} }$$

$$\displaystyle{ \cos^2(t) = \frac{1+\cos(2t)}{2} }$$

Trig Derivatives

 $$\displaystyle{ \frac{d[\sin(t)]}{dt} = \cos(t) }$$ $$\displaystyle{ \frac{d[\cos(t)]}{dt} = -\sin(t) }$$ $$\displaystyle{ \frac{d[\tan(t)]}{dt} = \sec^2(t) }$$ $$\displaystyle{ \frac{d[\cot(t)]}{dt} = -\csc^2(t) }$$ $$\displaystyle{ \frac{d[\sec(t)]}{dt} = \sec(t)\tan(t) }$$ $$\displaystyle{ \frac{d[\csc(t)]}{dt} = -\csc(t)\cot(t) }$$

Inverse Trig Derivatives

 $$\displaystyle{ \frac{d[\arcsin(t)]}{dt} = \frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arccos(t)]}{dt} = -\frac{1}{\sqrt{1-t^2}} }$$ $$\displaystyle{ \frac{d[\arctan(t)]}{dt} = \frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arccot(t)]}{dt} = -\frac{1}{1+t^2} }$$ $$\displaystyle{ \frac{d[\arcsec(t)]}{dt} = \frac{1}{\abs{t}\sqrt{t^2 -1}} }$$ $$\displaystyle{ \frac{d[\arccsc(t)]}{dt} = -\frac{1}{\abs{t}\sqrt{t^2 -1}} }$$

Trig Integrals

 $$\int{\sin(x)~dx} = -\cos(x)+C$$ $$\int{\cos(x)~dx} = \sin(x)+C$$ $$\int{\tan(x)~dx} = -\ln\abs{\cos(x)}+C$$ $$\int{\cot(x)~dx} = \ln\abs{\sin(x)}+C$$ $$\int{\sec(x)~dx} =$$ $$\ln\abs{\sec(x)+\tan(x)}+C$$ $$\int{\csc(x)~dx} =$$ $$-\ln\abs{\csc(x)+\cot(x)}+C$$

### Topics Listed Alphabetically

Single Variable Calculus

Multi-Variable Calculus

Differential Equations

Precalculus

### Search Practice Problems

Do you have a practice problem number but do not know on which page it is found? If so, enter the number below and click 'page' to go to the page on which it is found or click 'practice' to be taken to the practice problem.

 The 17Calculus and 17Precalculus iOS and Android apps are no longer available for download. If you are still using a previously downloaded app, your app will be available until the end of 2020, after which the information may no longer be available. However, do not despair. All the information (and more) is now available on 17calculus.com for free.

When using the material on this site, check with your instructor to see what they require. Their requirements come first, so make sure your notation and work follow their specifications.

DISCLAIMER - 17Calculus owners and contributors are not responsible for how the material, videos, practice problems, exams, links or anything on this site are used or how they affect the grades or projects of any individual or organization. We have worked, to the best of our ability, to ensure accurate and correct information on each page and solutions to practice problems and exams. However, we do not guarantee 100% accuracy. It is each individual's responsibility to verify correctness and to determine what different instructors and organizations expect. How each person chooses to use the material on this site is up to that person as well as the responsibility for how it impacts grades, projects and understanding of calculus, math or any other subject. In short, use this site wisely by questioning and verifying everything. If you see something that is incorrect, contact us right away so that we can correct it.