   Chapter 4.R, Problem 51E

Chapter
Section
Textbook Problem

# If f is continuous and ∫ 0 2 f ( x ) d x = 6 , evaluate ∫ 0 π / 2 f ( 2 sin θ )  cos θ   d θ .

To determine

To evaluate:

0π/2f2 sinθcosθ dθ

Explanation

1) Given:

02fxdx=6

2) Calculation:

The given integral is

0π/2f2 sinθcosθ dθ

Here use the substitution method

Thus substitute 2sinθ=u, therefore  2cosθdθ=du,  cosθdθ=du2

And the limits changes; the new limits of integration are calculated by substituting

For θ=0, u=0 & for θ=π2,u=2

Therefore, the given integral becomes

0π2f2 sinθcosθ dθ=02f(u)

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