   Chapter 4.5, Problem 3E

Chapter
Section
Textbook Problem

# Evaluate the integral by making the given substitution. ∫ x 2 x 3 + 1 d x ,    u = x 3 + 1

To determine

To evaluate:

The integral x2x3+1 dx by making the given substitution u=x3+1.

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:

x2x3+1 dx     u=x3+1

3) Calculation:

Use thesubstitution u=x3+1.

Differentiate u=x3+1 with respect to x.

du=3x2dx

As x2 dx is a part of the integration, solving for x2 dx by dividing both side by 3.

du3=x2dx

By using concept i)

Substitute u=x3+1, du3=x2dx

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