Trig Integration For Calculus 1
This page is for calculus 1 students who are just starting to learn trig integration. If you are in calculus 2 and you have mastered the basic trig integration techniques, you can go to this other page to learn some more advanced trig integration techniques.
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.
Recommended Books on Amazon (affiliate links)  

Basic Trigonometric Identities
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  doubleangle formulas  

\( \sin(2t) = 2\sin(t)\cos(t)\) 
\(\displaystyle{ \cos(2t) = \cos^2(t)  \sin^2(t) }\) 
Set 4  halfangle 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. Here are some practice problems.
Practice
Evaluate these integrals giving your answers in exact, simplified and factored form.
\(\displaystyle{ \int{ \cos(2x) ~dx } }\)
Problem Statement
Evaluate \(\displaystyle{ \int{ \cos(2x) ~dx } }\) giving your answer in exact, simplified, factored form.
Solution
video by The Organic Chemistry Tutor 

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

Evaluate \(\displaystyle{ \int_{\pi}^{\pi}{ \sin^2(nt) ~ dt } }\) where \(n\) is a positive integer.
Problem Statement 

Evaluate \(\displaystyle{ \int_{\pi}^{\pi}{ \sin^2(nt) ~ dt } }\) where \(n\) is a positive integer.
Final Answer 

\(\displaystyle{ \int_{\pi}^{\pi}{ \sin^2(nt) ~ dt } }\) \( = \pi \)
Problem Statement
Evaluate \(\displaystyle{ \int_{\pi}^{\pi}{ \sin^2(nt) ~ dt } }\) where \(n\) is a positive integer.
Solution
\(\displaystyle{ \int_{\pi}^{\pi}{ \sin^2(nt) ~ dt } }\) 
\(\displaystyle{ \int_{\pi}^{\pi}{ \frac{1\cos(2nt)}{2} ~ dt } }\) 
\(\displaystyle{ \frac{1}{2} \int_{\pi}^{\pi}{ 1\cos(2nt) ~ dt } }\) 
\(\displaystyle{ \frac{1}{2}\left[ t  \frac{\sin(2nt)}{2n} \right]_{\pi}^{\pi} }\) 
\(\displaystyle{ \frac{1}{2}\left[ \pi  \frac{\sin(2\pi n)}{2n} \right]  \frac{1}{2}\left[ \pi  \frac{\sin(2\pi n)}{2n} \right] }\) 
\(\sin(2\pi n) = 0\) and \(\sin(2\pi n) = 0\) for all \(n\) 
\(\displaystyle{ \frac{1}{2}\left[ \pi  \frac{0}{2n} \right]  \frac{1}{2}\left[ \pi  \frac{0}{2n} \right] }\) 
\(\displaystyle{ \pi/2 + \pi/2 }\) 
\(\displaystyle{ \pi }\) 
Final Answer
\(\displaystyle{ \int_{\pi}^{\pi}{ \sin^2(nt) ~ dt } }\) \( = \pi \)
Log in to rate this practice problem and to see it's current rating. 

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

Evaluate \(\displaystyle{ \int{ x\cos(x^2+1)~dx } }\) giving your answer in exact, simplified, factored form.
Final Answer 

\( (1/2)\sin(x^2+1) + C \)
Problem Statement
Evaluate \(\displaystyle{ \int{ x\cos(x^2+1)~dx } }\) giving your answer in exact, simplified, factored form.
Solution
video by Krista King Math 

Final Answer
\( (1/2)\sin(x^2+1) + C \)
Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \int{ 3(x^5)\sin(x^6)~dx } }\)
Problem Statement
Evaluate \(\displaystyle{ \int{ 3(x^5)\sin(x^6)~dx } }\) giving your answer in exact, simplified, factored form.
Solution
video by PatrickJMT 

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

\(\displaystyle{ \int{ \frac{\sin\sqrt{x}}{\sqrt{x}}~dx } }\)
Problem Statement
Evaluate \(\displaystyle{ \int{ \frac{\sin\sqrt{x}}{\sqrt{x}}~dx } }\) giving your answer in exact, simplified, factored form.
Solution
video by PatrickJMT 

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

\(\displaystyle{ \int{ 2\sin(x)\cos(x)~dx } }\)
Problem Statement
Evaluate \(\displaystyle{ \int{ 2\sin(x)\cos(x)~dx } }\) giving your answer in exact, simplified, factored form.
Solution
The first video solves the given problem. The second video, by a different instructor, evaluates the integral \(\int{\sin x \cos x ~ dx}\), which is the same as the given problem without the factor of 2. Consequently the answer in the second video is (1/2) times the answer in the first video.
video by Krista King Math 

video by The Organic Chemistry Tutor 

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

\(\displaystyle{ \int{ \sin^3(x) \cos(x) ~dx } }\)
Problem Statement
Evaluate \(\displaystyle{ \int{ \sin^3(x) \cos(x) ~dx } }\) giving your answer in exact, simplified, factored form.
Solution
video by Krista King Math 

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

\(\displaystyle{ \int{ \frac{\sin x}{\cos^4 x} ~ dx } }\)
Problem Statement
\(\displaystyle{ \int{ \frac{\sin x}{\cos^4 x} ~ dx } }\)
Solution
video by Integrals ForYou 

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

\(\displaystyle{ \int{ 5 \cos^4(2x) \sin(2x) ~dx } }\)
Problem Statement
Evaluate \(\displaystyle{ \int{ 5 \cos^4(2x) \sin(2x) ~dx } }\) giving your answer in exact, simplified, factored form.
Solution
video by PatrickJMT 

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

\(\displaystyle{ \int{ \frac{\cos^5(x) \sin(x)}{1  \sin^2(x)} dx } }\)
Problem Statement
Evaluate \(\displaystyle{ \int{ \frac{\cos^5(x) \sin(x)}{1  \sin^2(x)} dx } }\) giving your answer in exact, simplified, factored form.
Solution
video by PatrickJMT 

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

\(\displaystyle{ \int{ \frac{\cos(x) + \sin(x)}{\sin(2x)} ~dx } }\)
Problem Statement
Evaluate \(\displaystyle{ \int{ \frac{\cos(x) + \sin(x)}{\sin(2x)} ~dx } }\) giving your answer in exact, simplified and factored form.
Solution
video by PatrickJMT 

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

\(\displaystyle{ \int{ a\cos(x) + \frac{b}{\sin^2(x)} dx } }\)
Problem Statement
Evaluate \(\displaystyle{ \int{ a\cos(x) + \frac{b}{\sin^2(x)} dx } }\) giving your answer in exact, simplified, factored form.
Solution
video by Krista King Math 

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

\(\displaystyle{ \int{ \sin(2x) \cos(3x) ~dx } }\)
Problem Statement
Evaluate \(\displaystyle{ \int{ \sin(2x) \cos(3x) ~dx } }\) giving your answer in exact, simplified, factored form.
Solution
video by MIT OCW 

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

\(\displaystyle{ \int{ \frac{\sqrt[3]{\cot(x)}}{\sin^2(x)} ~dx } }\)
Problem Statement
Evaluate \(\displaystyle{ \int{ \frac{\sqrt[3]{\cot(x)}}{\sin^2(x)} ~dx } }\) giving your answer in exact, simplified and factored form.
Solution
video by PatrickJMT 

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

\(\displaystyle{ \int{ \tan^3(x) (\csc^2(x)1) ~dx } }\)
Problem Statement
Evaluate \(\displaystyle{ \int{ \tan^3(x) (\csc^2(x)1) ~dx } }\) giving your answer in exact, simplified, factored form.
Solution
video by PatrickJMT 

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

\(\displaystyle{ \int{ \csc x \cot x \sqrt{1\csc x}~dx } }\)
Problem Statement 

Evaluate \(\displaystyle{ \int{ \csc x \cot x \sqrt{1\csc x}~dx } }\) giving your answer in exact, simplified, factored form.
Final Answer 

\( (2/3)(1  \csc x)^{3/2} + C \)
Problem Statement
Evaluate \(\displaystyle{ \int{ \csc x \cot x \sqrt{1\csc x}~dx } }\) giving your answer in exact, simplified, factored form.
Solution
video by PatrickJMT 

Final Answer
\( (2/3)(1  \csc x)^{3/2} + C \)
Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \int{ \tan(x) \ln(\cos x) ~dx } }\)
Problem Statement
\(\displaystyle{ \int{ \tan(x) \ln(\cos x) ~dx } }\)
Solution
video by blackpenredpen 

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

\(\displaystyle{ \int_0^{\pi/2}{ \frac{1}{\cot x + \tan x} ~dx } }\)
Problem Statement 

Evaluate \(\displaystyle{ \int_0^{\pi/2}{ \frac{1}{\cot x + \tan x} ~dx } }\) giving your answer in exact, simplified and factored form.
Final Answer 

\(\displaystyle{ \int_0^{\pi/2}{ \frac{1}{\cot x + \tan x} ~dx } }\) \( = 1/2\)
Problem Statement
Evaluate \(\displaystyle{ \int_0^{\pi/2}{ \frac{1}{\cot x + \tan x} ~dx } }\) giving your answer in exact, simplified and factored form.
Solution
\(\displaystyle{ \int{ \frac{1}{\cos x/\sin x + \sin x/\cos x} ~dx } }\) 
\(\displaystyle{ \int{ \frac{1}{(\cos^2x + \sin^2 x)/(\sin x \cos x)} ~dx } }\) 
\(\displaystyle{ \int{ \frac{\sin x \cos x}{(\cos^2x + \sin^2 x)} ~dx } }\) 
\(\displaystyle{ \int{ \frac{\sin x \cos x}{1} ~dx } }\) 
\( \int{ \sin x \cos x ~dx } \) 
\( u = \sin x \to du = \cos x ~ dx \) 
\( \int{ u ~ du } \) 
\( u^2 /2 \) 
\( [(\sin^2 x) / 2]_{0}^{\pi/2} \) 
\( (\sin^2 (\pi/2)) / 2  (\sin^2 (0))/2 = 1/2 \) 
For the substitution, you could have let \(u=\cos x\) giving you \((\cos^2 x)/2\) for the indefinite integral. When you evaluate this at the endpoints, the answer is the same.
Final Answer
\(\displaystyle{ \int_0^{\pi/2}{ \frac{1}{\cot x + \tan x} ~dx } }\) \( = 1/2\)
Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \int_{0}^{\pi/2}{ x \cos(x) ~dx } }\)
Problem Statement
Evaluate \(\displaystyle{ \int_{0}^{\pi/2}{ x \cos(x) ~dx } }\) giving your answer in exact, simplified, factored form.
Solution
This problem is solved in two consecutive videos.
video by Krista King Math 

video by Krista King Math 

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

\(\displaystyle{ \int{ \cos^2x ~\tan^3x ~dx } }\)
Problem Statement
Evaluate \(\displaystyle{ \int{ \cos^2x ~\tan^3x ~dx } }\) giving your answer in exact, simplified, factored form.
Solution
video by PatrickJMT 

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

\(\displaystyle{ \int{ \frac{1}{\sin x} ~dx } }\)
Problem Statement 

Evaluate the integral \(\displaystyle{ \int{ \frac{1}{\sin x} ~dx } }\). Give your answer in simplified, factored form.
Hint 

This problem requires substitution and trig integration to solve. This problem looks like it should be easy but it isn't. There are at least four different ways to evaluate this integral, none of which may be obvious.
1. The weierstrass substitution \(t=\tan(x/2)\)
2. substitution \(u=x/2\)
3. substitution \(u=(1+\cos x)/(\sin x)\) after multiplying the integrand by \(u/u\)
4. start with \(\int{ \csc x ~dx}\) and let \(u=\csc x  \cot x\) after multiplying by \(u/u\)
Problem Statement 

Evaluate the integral \(\displaystyle{ \int{ \frac{1}{\sin x} ~dx } }\). Give your answer in simplified, factored form.
Final Answer 

\(\displaystyle{ \int{ \frac{1}{\sin x} ~dx } }\) \( = \ln\abs{ \tan(x/2) } + C_1 \) \( = \ln\abs{ \csc x  \cot x } + C_2 \)
Problem Statement
Evaluate the integral \(\displaystyle{ \int{ \frac{1}{\sin x} ~dx } }\). Give your answer in simplified, factored form.
Hint
This problem requires substitution and trig integration to solve. This problem looks like it should be easy but it isn't. There are at least four different ways to evaluate this integral, none of which may be obvious.
1. The weierstrass substitution \(t=\tan(x/2)\)
2. substitution \(u=x/2\)
3. substitution \(u=(1+\cos x)/(\sin x)\) after multiplying the integrand by \(u/u\)
4. start with \(\int{ \csc x ~dx}\) and let \(u=\csc x  \cot x\) after multiplying by \(u/u\)
Solution
This problem is solved several different ways as listed in the hint above.
Comment On Notation  Although his final answer is correct, he has some incorrect notation during the course of his solution. Notice that he doesn't include his constant of integration until the very end. To make the entire solution precisely correct, he needs to include the constant of integration in the step right after he does the actual integration. This is required since he writes equal signs between his steps. (This would also be required if he implied each step is equal to the previous one.) So don't do this or you may lose points for your work. However, as usual, check with your instructor to see what they require.
video by Integrals ForYou 

Comment On Notation  Although his final answer is correct, he has some incorrect notation during the course of his solution. Notice that he doesn't include his constant of integration until the very end. To make the entire solution precisely correct, he needs to include the constant of integration in the step right after he does the actual integration. This is required since he writes equal signs between his steps. (This would also be required if he implied each step is equal to the previous one.) So don't do this or you may lose points for your work. However, as usual, check with your instructor to see what they require.
video by Integrals ForYou 

Comment On Notation  Although his final answer is correct, he has some incorrect notation during the course of his solution. Notice that he doesn't include his constant of integration until the very end. To make the entire solution precisely correct, he needs to include the constant of integration in the step right after he does the actual integration. This is required since he writes equal signs between his steps. (This would also be required if he implied each step is equal to the previous one.) So don't do this or you may lose points for your work. However, as usual, check with your instructor to see what they require.
video by Integrals ForYou 

video by PatrickJMT 

Final Answer
\(\displaystyle{ \int{ \frac{1}{\sin x} ~dx } }\) \( = \ln\abs{ \tan(x/2) } + C_1 \) \( = \ln\abs{ \csc x  \cot x } + C_2 \)
Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \int{ \frac{1}{\cos x} ~dx } }\)
Problem Statement 

Evaluate the integral \(\displaystyle{ \int{ \frac{1}{\cos x} ~dx } }\). Give your answer in simplified, factored form.
Hint 

This problem requires substitution and trig integration to solve. This looks like it should be easy but it isn't. There are at least three different ways to evaluate the integral, none of which may be obvious.
1. The weierstrass substitution \(t=\tan(x/2)\)
2. start with \(\int{ \sec x ~dx}\) and let \(u=\sec x + \tan x \) after multiplying by \(u/u\)
3. substitution \(u=1/\cos x + \tan x\) after multiplying the integrand by \(u/u\)
Problem Statement
Evaluate the integral \(\displaystyle{ \int{ \frac{1}{\cos x} ~dx } }\). Give your answer in simplified, factored form.
Hint
This problem requires substitution and trig integration to solve. This looks like it should be easy but it isn't. There are at least three different ways to evaluate the integral, none of which may be obvious.
1. The weierstrass substitution \(t=\tan(x/2)\)
2. start with \(\int{ \sec x ~dx}\) and let \(u=\sec x + \tan x \) after multiplying by \(u/u\)
3. substitution \(u=1/\cos x + \tan x\) after multiplying the integrand by \(u/u\)
Solution
This problem is solved several different ways as listed in the hint above. The last two videos are by two different instructors that both use technique 3.
Comment On Notation  Although his final answer is correct, he has some incorrect notation during the course of his solution. Notice that he doesn't include his constant of integration until the very end. To make the entire solution precisely correct, he needs to include the constant of integration in the step right after he does the actual integration. This is required since he writes equal signs between his steps. (This would also be required if he implied each step is equal to the previous one.) So don't do this or you may lose points for your work. However, as usual, check with your instructor to see what they require.
video by Integrals ForYou 

video by Michel vanBiezen 

Comment On Notation  Although his final answer is correct, he has some incorrect notation during the course of his solution. Notice that he doesn't include his constant of integration until the very end. To make the entire solution precisely correct, he needs to include the constant of integration in the step right after he does the actual integration. This is required since he writes equal signs between his steps. (This would also be required if he implied each step is equal to the previous one.) So don't do this or you may lose points for your work. However, as usual, check with your instructor to see what they require.
video by Integrals ForYou 

video by Michel vanBiezen 

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

\(\displaystyle{ \int{ \sqrt{x} \sec(x^{3/2}) \tan(x^{3/2}) ~dx } }\)
Problem Statement
Evaluate \(\displaystyle{ \int{ \sqrt{x} \sec(x^{3/2}) \tan(x^{3/2}) ~dx } }\) giving your answer in exact, simplified, factored form.
Solution
video by PatrickJMT 

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

\(\displaystyle{ \int{ t(1+2t^2)^2 \csc^2\left[ (1+2t^2)^3 \right]~dt } }\)
Problem Statement
Evaluate \(\displaystyle{ \int{ t(1+2t^2)^2 \csc^2\left[ (1+2t^2)^3 \right]~dt } }\) giving your answer in exact, simplified, factored form.
Solution
video by PatrickJMT 

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

\(\displaystyle{ \int{ \frac{\sin x}{1+\sin x} ~ dx } }\)
Problem Statement
Evaluate \(\displaystyle{ \int{ \frac{\sin x}{1+\sin x} ~ dx } }\) giving your answer in exact, simplified and factored form.
Solution
video by allaboutintegration 

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

\(\displaystyle{\int_{\pi/2}^{3\pi/2}{\sqrt{1+\sin\theta}~d\theta}}\)
Problem Statement 

Evaluate \(\displaystyle{\int_{\pi/2}^{3\pi/2}{\sqrt{1+\sin\theta}~d\theta}}\) giving your answer in exact, simplified, factored form.
Final Answer 

Problem Statement
Evaluate \(\displaystyle{\int_{\pi/2}^{3\pi/2}{\sqrt{1+\sin\theta}~d\theta}}\) giving your answer in exact, simplified, factored form.
Solution
Our first inclination might be to use integration by substitution, letting \( u=1+\sin \theta \). However, since \( du= \cos \theta d \theta \) and we don't have a \(\cos(\theta)\) outside the square root, this won't help us.
So we will do a trick. We will multiply the integrand by \(\cos \theta / \cos \theta\). This doesn't seem to help us since now we have
\(\displaystyle{ \int_{\pi /2}^{3 \pi /2}{\sqrt{1+\sin \theta} \frac{\cos \theta}{\cos \theta}~ d \theta } } \)
However, now we are going to replace the cosine in the denominator with something that WILL help us using the identity \( \sin^2 \theta + \cos ^2 \theta = 1 \). This is a very simple identity but very powerful when evaluating trig integrals. Let's solve for \(\cos \theta\) here.
\( \cos \theta = \pm \sqrt{1\sin^2 \theta} \)
Notice in this last equation we have '\( \pm \)' in front of the square root. This is important. In algebra, your teacher may have let you ignore it. But in calculus you can't. You have to carry it along or consciously choose positive or negative. In this problem, you need to choose the negative sign because the limits of integration tell you that you are integrating in the left half plane where cosine is negative. Okay, so let's see where we are now. We are going to drop the limits of integration, so that we don't have to carry them along. Then we will bring them back in after we have evaluated the integral.
\(\displaystyle{ \int{ \sqrt{1+\sin \theta} \frac{\cos \theta}{\cos \theta} ~d \theta } }\) 
\(\displaystyle{ \int{ \sqrt{1+\sin \theta} \frac{\cos \theta}{\sqrt{1\sin^2 \theta}} ~d \theta } }\) 
\(\displaystyle{ \int{ \frac{\sqrt{1+\sin \theta}\cos \theta}{\sqrt{1+\sin \theta} \sqrt{1\sin \theta} } ~d \theta } }\) 
\(\displaystyle{ \int{ \frac{\cos \theta}{\sqrt{1\sin \theta} } ~d \theta } }\) 
Now integration by substitution will work with \(u=1\sin \theta \to du = \cos \theta d \theta \) giving us
\(\displaystyle{ \int{ \frac{\cos \theta}{\sqrt{1\sin \theta}} ~ d \theta} }\) 
\(\displaystyle{ \int{ \frac{du}{u^{1/2}} } }\) 
\(\displaystyle{ \int{ u^{1/2}~du } }\) 
\(\displaystyle{ \frac{u^{1/2}}{1/2} }\) 
\(\displaystyle{ 2u^{1/2} = 2 (1\sin \theta)^{1/2} }\) 
Normally when we work an indefinite integral, we need to add \(+C\) for the unknown constant. However, we are leaving it off since we know that ultimately our problem is a definite integral. So, let's finish the problem.
\(\displaystyle{ \left. 2 (1\sin \theta)^{1/2} \right_{\pi/2}^{3\pi/2} }\) 
\(\displaystyle{ 2(1  \sin(3\pi/2))^{1/2}  2(1  \sin(\pi/2))^{1/2} }\) 
\(\displaystyle{ 2(1(1))^{1/2}  2(11)^{1/2} = 2\sqrt{2} }\) 
Final Answer
Log in to rate this practice problem and to see it's current rating. 

\(\displaystyle{ \int{ \sqrt{\tan x} + \sqrt{\cot x} ~ dx } }\)
Problem Statement 

Use integration by parts to evaluate \(\displaystyle{ \int{ \sqrt{\tan x} + \sqrt{\cot x} ~ dx } }\) giving your answer in exact, simplified, factored form.
Hint 

Try to evaluate this integral with as few hints as possible. I am giving you progressive hints here that I think will help you get unstuck as you work this problem.
1. Convert both terms to sines and cosines.
2. Get a common denominator and combine into one term.
3. On the side expand out \( (\sin x  \cos x)^2 \) and see if you can use this in the denominator.
4. Use the substitution \( u = \sin x  \cos x \).
Problem Statement
Use integration by parts to evaluate \(\displaystyle{ \int{ \sqrt{\tan x} + \sqrt{\cot x} ~ dx } }\) giving your answer in exact, simplified, factored form.
Hint
Try to evaluate this integral with as few hints as possible. I am giving you progressive hints here that I think will help you get unstuck as you work this problem.
1. Convert both terms to sines and cosines.
2. Get a common denominator and combine into one term.
3. On the side expand out \( (\sin x  \cos x)^2 \) and see if you can use this in the denominator.
4. Use the substitution \( u = \sin x  \cos x \).
Solution
video by Integrals ForYou 

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

Really UNDERSTAND Calculus
Log in to rate this page and to see it's current rating.
To bookmark this page and practice problems, log in to your account or set up a free account.
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.
 
I recently started a Patreon account to help defray the expenses associated with this site. To keep this site free, please consider supporting me. 

Support 17Calculus on Patreon 
next: weierstrass substitution → 



Practice Instructions
Evaluate these integrals giving your answers in exact, simplified and factored form.