## 17Calculus Integrals - Trig Integration For Calculus 1

##### 17Calculus

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.

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 - 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.   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

### The Organic Chemistry Tutor - 2584 video solution

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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.

$$\displaystyle{ \int_{-\pi}^{\pi}{ \sin^2(nt) ~ dt } }$$ $$= \pi$$

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 $$n$$ $$\displaystyle{ \frac{1}{2}\left[ \pi - \frac{0}{2n} \right] - \frac{1}{2}\left[ -\pi - \frac{0}{2n} \right] }$$ $$\displaystyle{ \pi/2 + \pi/2 }$$ $$\displaystyle{ \pi }$$

$$\displaystyle{ \int_{-\pi}^{\pi}{ \sin^2(nt) ~ dt } }$$ $$= \pi$$

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

Problem Statement

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

$$(1/2)\sin(x^2+1) + C$$

Problem Statement

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

Solution

### Krista King Math - 88 video solution

video by Krista King Math

$$(1/2)\sin(x^2+1) + C$$

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

Problem Statement

Evaluate $$\displaystyle{ \int{ 3(x^5)\sin(x^6)~dx } }$$ giving your answer in exact, simplified, factored form.

Solution

### PatrickJMT - 1324 video solution

video by PatrickJMT

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

Problem Statement

Evaluate $$\displaystyle{ \int{ \frac{\sin\sqrt{x}}{\sqrt{x}}~dx } }$$ giving your answer in exact, simplified, factored form.

Solution

### PatrickJMT - 1325 video solution

video by PatrickJMT

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

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.

### Krista King Math - 90 video solution

video by Krista King Math

### The Organic Chemistry Tutor - 90 video solution

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

Problem Statement

Evaluate $$\displaystyle{ \int{ \sin^3(x) \cos(x) ~dx } }$$ giving your answer in exact, simplified, factored form.

Solution

### Krista King Math - 89 video solution

video by Krista King Math

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$$\displaystyle{ \int{ \frac{\sin x}{\cos^4 x} ~ dx } }$$

Problem Statement

$$\displaystyle{ \int{ \frac{\sin x}{\cos^4 x} ~ dx } }$$

Solution

### Integrals ForYou - 4416 video solution

video by Integrals ForYou

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

Problem Statement

Evaluate $$\displaystyle{ \int{ 5 \cos^4(2x) \sin(2x) ~dx } }$$ giving your answer in exact, simplified, factored form.

Solution

### PatrickJMT - 1330 video solution

video by PatrickJMT

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

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

### PatrickJMT - 105 video solution

video by PatrickJMT

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

Problem Statement

Evaluate $$\displaystyle{ \int{ \frac{\cos(x) + \sin(x)}{\sin(2x)} ~dx } }$$ giving your answer in exact, simplified and factored form.

Solution

### PatrickJMT - 101 video solution

video by PatrickJMT

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

Problem Statement

Evaluate $$\displaystyle{ \int{ a\cos(x) + \frac{b}{\sin^2(x)} dx } }$$ giving your answer in exact, simplified, factored form.

Solution

### Krista King Math - 83 video solution

video by Krista King Math

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

Problem Statement

Evaluate $$\displaystyle{ \int{ \sin(2x) \cos(3x) ~dx } }$$ giving your answer in exact, simplified, factored form.

Solution

### MIT OCW - 109 video solution

video by MIT OCW

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

Problem Statement

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

Solution

### PatrickJMT - 102 video solution

video by PatrickJMT

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

Problem Statement

Evaluate $$\displaystyle{ \int{ \tan^3(x) (\csc^2(x)-1) ~dx } }$$ giving your answer in exact, simplified, factored form.

Solution

### PatrickJMT - 106 video solution

video by PatrickJMT

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

Problem Statement

Evaluate $$\displaystyle{ \int{ \csc x \cot x \sqrt{1-\csc x}~dx } }$$ giving your answer in exact, simplified, factored form.

$$(2/3)(1 - \csc x)^{3/2} + C$$

Problem Statement

Evaluate $$\displaystyle{ \int{ \csc x \cot x \sqrt{1-\csc x}~dx } }$$ giving your answer in exact, simplified, factored form.

Solution

### PatrickJMT - 2111 video solution

video by PatrickJMT

$$(2/3)(1 - \csc x)^{3/2} + C$$

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$$\displaystyle{ \int{ \tan(x) \ln(\cos x) ~dx } }$$

Problem Statement

$$\displaystyle{ \int{ \tan(x) \ln(\cos x) ~dx } }$$

Solution

### blackpenredpen - 3562 video solution

video by blackpenredpen

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$$\displaystyle{ \int_0^{\pi/2}{ \frac{1}{\cot x + \tan x} ~dx } }$$

Problem Statement

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

$$\displaystyle{ \int_0^{\pi/2}{ \frac{1}{\cot x + \tan x} ~dx } }$$ $$= 1/2$$

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.

$$\displaystyle{ \int_0^{\pi/2}{ \frac{1}{\cot x + \tan x} ~dx } }$$ $$= 1/2$$

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

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.

### Krista King Math - 113 video solution

video by Krista King Math

### Krista King Math - 113 video solution

video by Krista King Math

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

Problem Statement

Evaluate $$\displaystyle{ \int{ \cos^2x ~\tan^3x ~dx } }$$ giving your answer in exact, simplified, factored form.

Solution

### PatrickJMT - 117 video solution

video by PatrickJMT

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$$\displaystyle{ \int{ \frac{1}{\sin x} ~dx } }$$

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.

$$\displaystyle{ \int{ \frac{1}{\sin x} ~dx } }$$ $$= \ln\abs{ \tan(x/2) } + C_1$$ $$= \ln\abs{ \csc x - \cot x } + C_2$$

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.

### Integrals ForYou - 4280 video solution

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

### Integrals ForYou - 4280 video solution

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

### Integrals ForYou - 4280 video solution

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

### PatrickJMT - 4280 video solution

video by PatrickJMT

$$\displaystyle{ \int{ \frac{1}{\sin x} ~dx } }$$ $$= \ln\abs{ \tan(x/2) } + C_1$$ $$= \ln\abs{ \csc x - \cot x } + C_2$$

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$$\displaystyle{ \int{ \frac{1}{\cos x} ~dx } }$$

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.

### Integrals ForYou - 4286 video solution

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

### Michel vanBiezen - 4286 video solution

video by Michel vanBiezen

### Integrals ForYou - 4286 video solution

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

### Michel vanBiezen - 4286 video solution

video by Michel vanBiezen

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

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

### PatrickJMT - 1326 video solution

video by PatrickJMT

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

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

### PatrickJMT - 1327 video solution

video by PatrickJMT

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$$\displaystyle{ \int{ \frac{\sin x}{1+\sin x} ~ dx } }$$

Problem Statement

Evaluate $$\displaystyle{ \int{ \frac{\sin x}{1+\sin x} ~ dx } }$$ giving your answer in exact, simplified and factored form.

Solution

### allaboutintegration - 3860 video solution

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

Problem Statement

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

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

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.

 $$\displaystyle{ \left. 2 (1-\sin \theta)^{1/2} \right|_{\pi/2}^{3\pi/2} }$$ $$\displaystyle{ 2(1 - \sin(3\pi/2))^{1/2} - 2(1 - \sin(\pi/2))^{1/2} }$$ $$\displaystyle{ 2(1-(-1))^{1/2} - 2(1-1)^{1/2} = 2\sqrt{2} }$$

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

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$$\displaystyle{ \int{ \sqrt{\tan x} + \sqrt{\cot x} ~ dx } }$$

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

### Integrals ForYou - 3805 video solution

video by Integrals ForYou

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