   Chapter 4.5, Problem 21E

Chapter
Section
Textbook Problem

# Evaluate the indefinite integral. ∫ a + b 2 3 a x + b x 3 d x

To determine

To evaluate:

The indefiniteintegral a+bx23ax+bx3dx

Explanation

1) Concept:

i) The substitution rule

If u=g(x) is a differentiable function whose range is an interval I and f is continuous on I, then f(gx)g'xdx=f(u)du.

ii) Indefinite integral

xn dx=xn+1n+1+C  (n-1)

2) Given:

a+bx23ax+bx3dx

3) Calculation:

Here, use the substitution method because the differential of the function 3ax+bx3 is

(3a+3bx2)dx.

Substitute u=3ax+bx3.

Differentiate u=3ax+bx3 with respect to x.

du=(3a+3bx2)dx

Factoring out 3 common,

du=3(a+bx2)dx

As (a+bx2)dx is a part of the integration, solving for (a+bx2)dx by dividing both side by 3

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