   Chapter 4.5, Problem 82E

Chapter
Section
Textbook Problem

# The following exercise are intended only for these who have already covered Chapter 6.Evaluate the integral. ∫ 0 1 x e − x 2   d x

To determine

To evaluate:

The given definite integral.

Explanation

1) Concept:

i) The substitution rule: for definite integral: If g'(x) is a continuous function on a,b whose f is continuous on range of u=g(x), then abfgxg'(x)dx=g(a)g(b)f(u)du. Here g(x) is substituted as u and then g(x)dx =du

ii)

xndx=xn+1n+1 ( n  -1)

2) Given:

01xe-x2dx

3) Calculation:

The given integral is

01xe-x2dx

Here using the substitution method

Substitute -x2=u

Differentiating with respect to x

-2xdx=du

xdx=-12du

The limits changes; the new limits of integration are calculated by substituting

For x=0,

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