   Chapter 4.5, Problem 55E

Chapter
Section
Textbook Problem

Change of Variables In Exercises 53-60, find the indefinite integral by making a change of variables. ∫ x 2 1 − x   d x

To determine

To calculate: The indefinite integral x21xdx.

Explanation

Given:

The provided integral is:

x21xdx

Formula used:

The power rule for integration is:

xndx=xn+1n+1+c,  n1

The algebric expression:

(ab)2=a22ab+b2

Calculation:

According to theorem for change of variable for indefinite integrals

If u=g(x) then du=g'(x)

Then integral will take the following form

f(g(x))g'(x)dx=f(u)du

Consider u=1x. Therefore x=1u.

Square the equation x=1u as below:

x2=(1u)2=u22u+1

Differentiate the equation u=1x with respect to x:

dudx=01du=dx

Put the values 1x=u, x=1u and du=dx in the provided integral:

x2x1dx=(u22u+1)udu=(u522u32+u)du=(u52du2u32du+udu

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