   Chapter 4.5, Problem 21E

Chapter
Section
Textbook Problem

Finding an Indefinite Integral In Exercises 9-30, find the indefinite integral and check the result by differentiation. ∫ 7 x ( 1 − x 2 ) 3 d x

To determine

To calculate: The indefinite integral 7x(1x2)3dx and verify it by differentiation.

Explanation

Given:

The integral 7x(1x2)3dx.

Formula used:

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=F(u)+C

Calculation: Consider, u=1x2.

Then, differentiate above equation with respect to x:

dudx=(02x)du=(2x)dxdu2=xdx

Multiply both sides by 7.

7du2=xdx

So, convert integral in terms of u,

7x(1x2)3dx=(72)1u3du

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