: Use the result ofExample 3 to evaluate'피/2dx1 + sin x

To evaluate the integral ∫0π2dx1+sinx Solution: The given integral is, ∫0π2dx1+sinx Multiplying…

Answered: : Use the result of Example 3 to…

College Algebra (MindTap Course List)

12th Edition

ISBN:9781305652231

Author:R. David Gustafson, Jeff Hughes

Publisher:R. David Gustafson, Jeff Hughes

Chapter3: Functions

Section3.5: Inverse Functions

Problem 1SC: Determine whether fx=x is one-to-one.

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Related questions

Question

: Use the result of
Example 3 to evaluate
'피/2
dx
1 + sin x

Expert Solution

Step 1

To evaluate the integral $\int_{0}^{\frac{π}{2}} \frac{d x}{1 + \sin x}$

Solution:

The given integral is,

$\int_{0}^{\frac{π}{2}} \frac{d x}{1 + \sin x}$

Multiplying both sides by $(1 - \sin x)$ as,

$\begin{array}{rcl} \int_{0}^{\frac{π}{2}} \frac{(1 - \sin x) d x}{(1 + \sin x) (1 - \sin x)} & = & \int_{0}^{\frac{π}{2}} \frac{(1 - \sin x) d x}{(1 - \sin^{2} x)} \\ = & \int_{0}^{\frac{π}{2}} \frac{1 - \sin x}{\cos^{2} x} d x \\ = & \int_{0}^{\frac{π}{2}} (s e c^{2} x - \tan x s e c x) d x \end{array}$