   Chapter 8.2, Problem 52E

Chapter
Section
Textbook Problem

Evaluating a Definite Integral In Exercises 43–52, evaluate the definite integral. Use a graphing utility to confirm your result. ∫ 0 π / 8 x ( sec 2 2 x ) d x

To determine

To calculate: The value of definite integral 0π/8x(sec22x)dx and also verify the result by using a graphing utility tool.

Explanation

Given:

The given definite integral is 0π/8x(sec22x)dx

Formula Used: by using formula of integration which is given as,

xn=xn+1n+1

We have,

(sec22x)dx=12tan2x

and

tan2xdx=12ln(cos2x)

On differentiating, we get,

ddxarcsecx=1xx21

Calculation:

Let

u=x

and

dv=(sec22x)dx ……(I)

On Integrating equation (I),

We get,

dv=(sec22x)dx

dv=12tan2x+C where, C is a constant. ……(II)

0π/8x(sec22x)dx=x0π/8(sec22x)dx0π/8(ddxx(0π/8(sec22x)dx))dx

Now, by using Integration by parts method, we have

0π/8x(sec

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