Evaluate the indefinite integral.∫sinθ(1+sinθ)84cosθdθ\int^{}sin\theta\left(1+\sin\theta\right)^{84}\cos\theta d\theta

1 86 ( 1 + sin ⁡ θ ) 86 − 1 85 ( 1 + sin ⁡ θ ) 85 + C \frac{1}{86}\left(1+\sin\theta\right)^{86}-\frac{1}{85}\left(1+\sin\theta\right)^{85}+C

Evaluate the indefinite integral. ∫ s i n θ ( 1 + sin θ ) 84 cos θ d θ \int^{}sin\theta\left(1+\sin\theta\right)^{84}\cos\theta d\theta

Here we're asked to evaluate the indefinite integral of the sine of theta times one plus the sine of theta to the power of 84 times the cosine of theta d e theta. Now this integral can look a little bit intimidating right off the bat, but we can attack it the same way we have all of our integrals that we've been working on using substitution. So getting started with our very first step here, we wanna choose u and find du. Now remember that u is often going to be a function inside of something either in parentheses or in an exponent or under a radical and it can be kind of tricky to make that decision when working with trig functions. Now in this case, we see that we have this function one plus the sine of theta raised to the eighty fourth power. So because it is inside of parentheses and raised to a power, that is a good clue that that would be a good choice for you. So let's start there. If u is equal to one plus sine of theta, what then is du? Well, du is going to be equal to the derivative of one plus sine of theta. The derivative of one is zero. The derivative of sine theta is cosine theta. So this is cosine of theta times d theta four du. So we're looking pretty good so far, but let's go ahead and move on to step two, rewriting our integral only in terms of u and du because this is often where things get a little bit tricky. We can see in our integral that this one plus sine theta is what we chose as u. Then du was cosine of theta d theta, which we see is conveniently right here in our integral. But we still have this extra sine of theta going on. So how exactly can we deal with that? Well, in the past, we've seen extra variables. Here, we act actually have an extra function of a variable. So we can actually use the same method that we did before. We can rewrite x or in this case, we can actually rewrite the sine of theta in terms of u. Because we chose u to be one plus the sine of theta. If I go ahead and subtract one from both sides, that then means that the sine of theta is equal to u minus one. So if sine of theta is equal to u minus one, I can go ahead and replace that in my integral. So that then makes this integral u minus one times u to the power of 84 du. So now we have an integral that looks pretty easy to work with. We can go ahead and move on to step three. We already rewrote our integral in terms of only u and du, and now we can integrate with respect to you. I'm gonna go ahead and distribute this u to the power of 84 into my parentheses here. This will give me a u to the power of 85 because remember, u to the power of one times u to the power of 84, that's u to the power of 85 and then minus u to the power of 84 du. Now we can just apply our power rule here, finding this antiderivative. So adding one to our exponent, dividing by that new exponent, this is going to give me one over 86 times u to the power of 86 and then minus applying our power rule again here. Add one to that exponent divide by that new exponent. That's one over 85 times u to the power of 85 and then plus our constant of integration c because this is an indefinite integral. So now that we're here, we have fully integrated this with respect to u. Now we wanna put this back in terms of our original variable, which in this case was theta. So we're gonna replace u with our original function of theta, which was one plus sine of theta. So we're gonna replace both of these u's with that. So this is gonna be one over 86 times one plus the sine of theta to the power of 86. I know I'm gonna need a bit more space here, so I'm gonna go ahead and move this over. Minus one over 85 times one plus sine theta to the power of 85 and then plus that constant of integration c. So this is my final answer here, having used substitution to evaluate what looked like a very scary integral at first. Feel free to let us know if you have any questions and I'll see you in the next one.

Evaluate the indefinite integral. ∫ s i n θ ( 1 + sin ⁡ θ ) 84 cos ⁡ θ d θ \int^{}sin\theta\left(1+\sin\theta\right)^{84}\cos\theta d\theta

1 86 ( 1 + sin ⁡ θ ) 86 − 1 85 ( 1 + sin ⁡ θ ) 85 + C \frac{1}{86}\left(1+\sin\theta\right)^{86}-\frac{1}{85}\left(1+\sin\theta\right)^{85}+C

− 1 85 ( sin ⁡ θ ) 85 − 1 86 ( sin ⁡ θ ) 86 -\frac{1}{85}\left(\sin\theta\right)^{85}-\frac{1}{86}\left(\sin\theta\right)^{86}

1 86 ( cos ⁡ θ ) 86 − 1 85 ( cos ⁡ θ ) 85 + C \frac{1}{86}\left(\cos\theta\right)^{86}-\frac{1}{85}\left(\cos\theta\right)^{85}+C

1 86 ( 1 + sin ⁡ θ ) 86 − 1 86 ( 1 + sin ⁡ θ ) 86 + C \frac{1}{86}\left(1+\sin\theta\right)^{86}-\frac{1}{86}\left(1+\sin\theta\right)^{86}+C

8. Definite Integrals

Substitution

Calculus

8. Definite Integrals

Substitution

Struggling with Calculus?

Join thousands of students who trust us to help them ace their exams!Watch the first video

Multiple Choice

Evaluate the indefinite integral.
$\int^{} s i n θ {(1 + \sin θ)}^{84} \cos θ d θ$

$- \frac{1}{85} {(\sin θ)}^{85} - \frac{1}{86} {(\sin θ)}^{86}$

$\frac{1}{86} {(\cos θ)}^{86} - \frac{1}{85} {(\cos θ)}^{85} + C$

$\frac{1}{86} {(1 + \sin θ)}^{86} - \frac{1}{85} {(1 + \sin θ)}^{85} + C$

$\frac{1}{86} {(1 + \sin θ)}^{86} - \frac{1}{86} {(1 + \sin θ)}^{86} + C$

Verified step by step guidance

Step 1: Recognize that the integral involves a composite function. The term $ (1 + \sin\theta)^{84} $ suggests that substitution might simplify the problem. Let $ u = 1 + \sin\theta $.

Step 2: Differentiate $ u $ with respect to $ \theta $. Since $ u = 1 + \sin\theta $, we have $ \frac{du}{d\theta} = \cos\theta $, or equivalently, $ du = \cos\theta \, d\theta $.

Step 3: Rewrite the integral in terms of $ u $. Substituting $ u = 1 + \sin\theta $ and $ du = \cos\theta \, d\theta $, the integral becomes $ \int \sin\theta \, u^{84} \, \cos\theta \, d\theta $. Note that $ \sin\theta = u - 1 $, so the integral becomes $ \int (u - 1) u^{84} \, du $.

Step 4: Simplify the integrand. Expand $ (u - 1) u^{84} $ to get $ u^{85} - u^{84} $. The integral now becomes $ \int u^{85} \, du - \int u^{84} \, du $.

Step 5: Integrate each term. Use the power rule for integration, $ \int u^n \, du = \frac{u^{n+1}}{n+1} + C $, to find the antiderivative of each term. After integrating, substitute back $ u = 1 + \sin\theta $ to express the result in terms of $ \theta $.

8:38m

Watch next

Master Indefinite Integrals with a bite sized video explanation from Patrick

Start learning

Substitution practice set

Problem sets built by lead tutors

Expert video explanations

Go to Practice