   Chapter 7.3, Problem 13E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ x 2 − 9 x 3   d x

To determine

To evaluate: The given integral x29x3dx.

Explanation

Integration involving terms of the form x2a2 can be simplified by using the trigonometric substitution x=asecθ.

Formula used:

The identity, tan2x=sec2x1

Given:

The integral, x29x3dx

Calculation:

Substitute for x as x=3secθ. Take the derivative of the substitution term:

x=3secθdx=3secθtanθdθ

Here, 0θ<π2

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

x29x3dx=9sec2θ933sec3θ3secθtanθdθ=3sec2θ133sec3θ3secθtanθdθ=13sec2θ1sec2θtanθdθ

Use the identity tan2x=sec2x1:

x29x3dx=13tan2θsec2θtanθdθ=13tanθsec2θtanθdθ=13tan2θsec2θdθ

In terms of sine and cosine, the integration will be:

x29x3dx=13tan2θsec2θdθ=13sin2θcos2θcos2θdθ

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