# Evaluate this integral using trigonometric identities. \int_{0}^{\frac{\pi}{2}} \sin^7{\theta}...

## Question:

Evaluate this integral using trigonometric identities.

{eq}\displaystyle \int_{0}^{\frac{\pi}{2}} \sin^7{\theta} \cos^{5}{\theta} d\theta {/eq}

## Integration using Substitution:

We determine the value of integrals using different methods and one of them is the method of substitution. This technique involves the substitution of a term in the integrand to another variable. With this substitution, the differential variable and the limits of integration also change. The integral is done on the transformed integrand.

## Answer and Explanation:

For the given function, we can apply the trigonometric identity, {eq}\displaystyle \cos^2 \theta = 1-\sin^2 \theta {/eq}. Afterwards, we proceed with the substitution on the integrand wherein we allow {eq}\displaystyle u = \sin\theta {/eq} which corresponds to a differential {eq}\displaystyle du = \cos\theta {/eq}, and a change in the limits of integration, {eq}\displaystyle 0\to \sin 0 = 0 {/eq} and {eq}\displaystyle \frac{\pi}{2}\to \sin \frac{\pi}{2} = 1 {/eq}. Now, we proceed with the integration through the following steps.

{eq}\begin{align} \displaystyle \int_{0}^{\frac{\pi}{2}} \sin^7\theta \cos^5\theta d\theta &= \int_0^\frac{\pi}{2} \sin^7\theta (1-\sin^2\theta)\cos\theta d\theta\\ &= \int_0^\frac{\pi}{2} ( \sin^7\theta-\sin^9\theta)\cos\theta d\theta\\ &= \int_0^1 (u^7 - u^9)du\\ &=\left( \frac{u^8}{8} - \frac{u^{10}}{10}\right) \bigg|_0^1\\ &= \frac{1}{8}(1^8-0^8) - \frac{1}{10}(1^{10}- 0^{10})\\ &= \frac{1}{8} - \frac{1}{10}\\ &= \frac{1}{40} \end{align} {/eq}

#### Learn more about this topic: How to Solve Integrals Using Substitution

from Math 104: Calculus

Chapter 13 / Lesson 5
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