#### To determine

**To evaluate:**

The given definite integral.

#### Answer

*121-1e*

#### 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, u=-02=0, and for x=1, u=(-)12=-1

Therefore, the given integral becomes

∫01xe-x2dx=∫0-1eu-12du

Taking the constant outside the integral

=-12∫0-1eudu

Integrating this by using the fundamental theorem of calculus and power rule of integration

*=-12eu0-1*

= -12e-1-e0

=-121e-1

=121-1e

**Conclusion:**

Therefore,

∫01xe-x2dx=121-1e