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You CAN Ace Calculus

17calculus > integrals > trig integration

for everyone

integration

integration by substitution

additional topics for people in group 2

integration by parts

### Calculus Main Topics

Integrals

Integral Applications

Single Variable Calculus

Multi-Variable Calculus

### Tools

math tools

general learning tools

Trig Integration - Integration of Trigonometric Functions

### Start Here

This page is directed to two main groups of students. To best use the material and resources on this page, determine which group you are in and follow these steps.

Group 1: If you are in first semester calculus and you are just learning integration, read the first few paragraphs of this page through the the section on basic integration. This will explain the material on this page that applies to you. In the practice problem section, filtering out all practice problems except for the basic trig integrals will display only the practice problems that apply to you. Most of the rest of this page and the other practice problems apply to people in group 2.

Group 2: If you are in second semester calculus and you have learned integration by parts, follow these steps.
Step 1: Read the panel immediately following this one discussing the difference between trig integration and trig substitution to make sure you are on the correct page. The material on this page is usually covered before the material on trig substitution.
Step 2: Go through first few paragraphs on this page through the basic integration section to refresh your memory, if you feel like you need to, and then read the rest of this page.
Step 3: Once you are ready to watch some videos and try some practice problems, go to the resources section below to learn and practice.

### Difference Between Trig Integration and Trig Substitution

Trig integration, covered on this page, is the evaluation of integrals that already have trig functions in the integrand.

trig substitution is a technique that takes an integrand that most likely does NOT contain any trig functions, and uses some trig identities to introduce trig functions into the integrand. Once the integral is completely transformed, then trig integration is used to evaluate the integral. Once the evaluation is complete, another set of substitutions, based on the original ones, is done to convert the result back to the original variable.

When integrating trig functions, the trick is to get the integrand into a form so that you can use integration by substitution. Besides the basic trig identities, there are several sets of identities that you need to know and be able to use.

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

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

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

Basic Trigonometric Integration

When you are first starting to learn integration, you will run across problems involving trig functions. The basic information and techniques you need are the basic trig identities and integration by substitution. In addition to the identities in the table above, you need to know these integrals.

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

With the equations in the previous two tables, you should have all the tools you need to solve basic trig integrals. The remaining material on this page is for people who are in group 2 (as described in the above Start Here panel).

Here are links to other pages involving more advanced techniques, usually found in second semester calculus.

The strategies listed so far on this page cover most of the integrals you will run across. Occasionally, you may need various other techniques to convert the integrand into a form that can be integrated. Here are a few ideas.
1. Convert all the trig functions into sine and cosine. Sometimes, you may have lots of cancellation and end up with an easy integral.
2. Use the half-angle formulas to remove powers. This may leave an easy substitution problem.

The rest of this page covers a unique, more specialized strategy that you can try. Once you get more experience, you will be able to tell pretty quickly which strategy is best.

 Basic Problems

Practice 1

$$\displaystyle{\int{x\cos(x^2+1)~dx}}$$

solution

Practice 2

$$\displaystyle{\int{3x^5\sin(x^6)~dx}}$$

solution

Practice 3

$$\displaystyle{\int{\frac{\sin\sqrt{x}}{\sqrt{x}}~dx}}$$

solution

Practice 4

$$\displaystyle{\int{2\sin(x)\cos(x)~dx}}$$

solution

Practice 5

$$\displaystyle{\int{\sin^3(x)\cos(x)~dx}}$$

solution

Practice 6

$$\displaystyle{\int{5\cos^4(2x)\sin(2x)~dx}}$$

solution

Practice 7

$$\displaystyle{ \int{\frac{\cos^5(x)\sin(x)}{1 - \sin^2(x)}dx} }$$

solution

Practice 8

$$\displaystyle{ \int{\frac{\cos(x) + \sin(x)}{\sin(2x)}~dx} }$$

solution

Practice 9

$$\displaystyle{\int{a\cos(x)+\frac{b}{\sin^2(x)}dx}}$$

solution

Practice 10

$$\displaystyle{\int{\sin(2x)\cos(3x)~dx}}$$

solution

Practice 11

$$\displaystyle{ \int{\frac{\sqrt[3]{\cot(x)}}{\sin^2(x)}~dx} }$$

solution

Practice 12

$$\displaystyle{\int{\tan^3(x)(\csc^2(x)-1)~dx}}$$

solution

Practice 13

$$\displaystyle{ \int{ \csc x \cot x \sqrt{1-\csc x}~dx } }$$

solution

 Intermediate Problems

Practice 14

$$\displaystyle{\int_{0}^{\pi/2}{x\cos(x)~dx}}$$

solution

Practice 15

$$\displaystyle{\int{\cos^2x~\tan^3x~dx}}$$

solution

Practice 16

$$\displaystyle{\int{\csc(x)~dx}}$$

solution

Practice 17

$$\displaystyle{ \int{\sqrt{x}\sec(x^{3/2})\tan(x^{3/2})~dx} }$$

solution

Practice 18

$$\displaystyle{ \int{(1+2t^2)^2~t~\csc^2\left[(1+2t^2)^3\right]~dt} }$$

solution

Practice 19

$$\displaystyle{\int_{\pi/2}^{3\pi/2}{\sqrt{1+\sin\theta}~d\theta}}$$    tricky!

hint

solution

Special Tangent Substitution

An interesting and special substitution that will often convert trig integrals into a form that can be integrated is to let $$t = \tan(x/2)$$. From this we get the list of substitutions in the table below.

 $$\displaystyle{ t = \tan(x/2) }$$ $$\displaystyle{ \sin(x) = \frac{2t}{1+t^2} }$$ $$\displaystyle{ \cos(x) = \frac{1-t^2}{1+t^2} }$$ $$\displaystyle{ dx = \frac{2}{1+t^2}dt }$$

This video shows the derivation of these equations. It is recommended that you watch it, so that you will know where the equations come from and how to use them.

 PatrickJMT - Integrate Rational Function of Sine and Cosine ; t = tan(x/2) , Part 1

Okay, let's work some practice problems using this substitution.

Practice 20

$$\displaystyle{\int{\frac{dx}{2\sin(x)+\sin(2x)}}}$$