   Chapter 7.3, Problem 29E

Chapter
Section
Textbook Problem

# Evaluate the integral. ∫ x 1 − x 4 d x

To determine

To evaluate: The given integral x1x4dx.

Explanation

Integration involving terms of the form a2x2 can be simplified by using the trigonometric substitution x=asinθ.

Formula used:

The identity, cos2x=1sin2x

and cos2x=1+cos2x2

Given:

The integral, x1x4dx

Calculation:

Let x2=t, then:

dt=2xdxxdx=12dt

Change the integral from x to t using the above substitution:

x1x4dx=1x4xdx=121t2dt

Now, substitute t=sinθ,du=cosθdθ, then the integration becomes:

x1x4dx=121sin2θcosθdθ

Here, π2θπ2. Simplify further using the identity cos2x=1sin2x:

x1x4dx=121sin2θcosθdθ=12cos2θcosθdθ=12cos2θdθ

Use the identity cos2x=1+cos2x2:

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