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


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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