   Chapter 7.3, Problem 22E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ 0 1 x 2 + 1 ​   d x

To determine

To evaluate: The given integral 01x2+1dx.

Explanation

Integration involving terms of the form a2+x2 can be simplified by using the trigonometric substitution x=atanθ.

Formula used:

The formula for integration by parts in terms of u and v is given by

udv=uvvdu

The identity, sec2x=1+tan2x

Given:

The integral, 01x2+1dx

Calculation:

Substitute for x as x=tanθ. Take the derivative of the substitution term:

x=tanθdx=sec2θdθ

Here, π2<θ<π2. The limits of integration will change as:

x0tanθ=0θ0andx1tanθ=1θπ4

Substitute for x and dx in the given integral to get:

01x2+1dx=0π4tan2θ+1sec2θdθ

Use the identity sec2x=1+tan2x:

01x2+1dx=0π4sec2θsec2θdθ=0π4sec3θdθ=0π4secθsec2θdθ

Solve the integration by method of integration by parts. Make the choice for u and dv such that the resulting integration from the formula above is easier to integrate. Let

u=secθ      dv=sec2θdθ

Then, the differentiation of u and antiderivative of dv will be

du=secθtanθdθ      v=tanθ

Substitute for variables in the formula above to get

0π4secθsec2θdθ=secθtanθ]0π40π4ta

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