   Chapter 4.5, Problem 19E

Chapter
Section
Textbook Problem

# Evaluate the indefinite integral. ∫ ( x 2 + 1 ) ( x 3 + 3 x ) 4 d x

To determine

To evaluate:

The indefinite integral x2+1x3+3x4 dx

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:

x2+1x3+3x4 dx

3) Calculation:

Here, use the substitution method because the differential of the function x3+3x is 3x2+3dx

Substitute u=x3+3x.

Differentiate u=x3+3x with respect to x.

du=3x2+3dx

Factoring out 3 common,

du=3x2+1dx

As x2+1dx is a part of the integration, solving for x2+1dx by dividing both side by 3

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