Evaluate the integral.∫0 π/2 Sin5 x dx

To evaluate the integral: ∫0π2sin5xdx Solution: Given integral can be written as:…

Answered: Evaluate the integral. ∫0 π/2 Sin5 x…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

Evaluate the integral.

∫₀ ^π/2Sin⁵ x dx

Expert Solution

Step 1

To evaluate the integral: $\int_{0}^{\frac{π}{2}} \sin^{5} x d x$

Solution:

Given integral can be written as:

$\int_{0}^{\frac{π}{2}} \sin^{5} x d x = \int_{0}^{\frac{π}{2}} \sin^{4} x \cdot \sin x d x$

We know that $\sin^{2} x = 1 - \cos^{2} x$ .

Using the above formula, integrand can be written as:

$\begin{array}{rcl} \int_{0}^{\frac{π}{2}} \sin^{5} x d x & = & \int_{0}^{\frac{π}{2}} \sin^{4} x \cdot \sin x d x \\ = & \int_{0}^{\frac{π}{2}} {(1 - \cos^{2} x)}^{2} \sin x d x \end{array}$

Step 2

Now, let $t = \cos x$

$\begin{array}{rcl} \frac{d t}{d x} & = & - \sin x \\ - d t & = & \sin x d x \end{array}$

Limit will change accordingly.

When $x \to \frac{π}{2}$ then $t \to 0$ .

When $x \to 0$ then $t \to 1$ .

$\begin{array}{rcl} \int_{0}^{\frac{π}{2}} {(1 - \cos^{2} x)}^{2} \sin x d x & = & \int_{1}^{0} {(1 - t^{2})}^{2} \cdot - d t \\ = & - \int_{1}^{0} (1 - 2 t^{2} + t^{4}) d t \\ = & - {[t - \frac{2 t^{3}}{3} + \frac{t^{5}}{5}]}_{1}^{0} \\ = & - [0 - (1 - \frac{2}{3} + \frac{1}{5})] \\ = & - [- (\frac{15 - 10 + 3}{15})] \\ = & \frac{8}{15} \end{array}$

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