   Chapter 4.5, Problem 60E

Chapter
Section
Textbook Problem

# If f is continuous and ∫ 0 9 f ( x ) d x = 4 , find ∫ 0 3 x f ( x 2 ) d x .

To determine

To find:

03xf(x2)dx

Explanation

1) Concept:

The substitution rule:

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

Here, g(x) is substituted as u and then g(x)dx =du.

2) Given:

The function f  is continuous and 09f(x)dx=4

3) calculation:

To find

03xf(x2)dx

use the substitution method.

Substitute x2=u.

Differentiating with respect to x

2xdx=du

xdx=12du

The limits change and the new limits of integration are calculated by substituting

for x=0,  u=02</

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