   Chapter 4.5, Problem 2E

Chapter
Section
Textbook Problem

# Evaluate the integral by making the given substitution. ∫ x ( 2 x 2 + 3 ) 4 d x ,    u = 2 x 2 + 3

To determine

To evaluate:

The integral x2x2+34 dx by making the given substitution u=2x2+3.

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:

x2x2+34 dx,     u=2x2+3

3) Calculation:

Use thesubstitution u=2x2+3.

Differentiate u=2x2+3 with respect to x.

du=4xdx

As x dx is a part of the integration, solving for x dx by dividing both sideby 4.

xdx=du4

By using concept i)

Substitute u=2x2+3, xdx=du4

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