Mister Exam

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How to use it?
Integral of d{x}:
Integral of sin^4(x)
Integral of x^-3
Integral of (sin(x)^2)/(cos(x)^4)
Integral of sint
Limit of the function:
x*exp(x^2/2)
Identical expressions
x*exp(x^ two / two)
x multiply by exponent of (x squared divide by 2)
x multiply by exponent of (x to the power of two divide by two)
x*exp(x2/2)
x*expx2/2
x*exp(x²/2)
x*exp(x to the power of 2/2)
xexp(x^2/2)
xexp(x2/2)
xexpx2/2
xexpx^2/2
x*exp(x^2 divide by 2)
x*exp(x^2/2)dx

Solving integrals/ x*exp(x^2/2)

Integral of x*exp(x^2/2) dx

The solution

You have entered [src]

  1         
  /         
 |          
 |      2   
 |     x    
 |     --   
 |     2    
 |  x*e   dx
 |          
/           
0

$$\int\limits_{0}^{1} x e^{\frac{x^{2}}{2}}\, dx$$

Integral(x*exp(x^2/2), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = e^{\frac{x^{2}}{2}}$ .
  
  Then let $du = x e^{\frac{x^{2}}{2}} dx$ and substitute $du$ :
  
  $\int 1\, du$
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 1\, du = u$
  Now substitute $u$ back in:
  
  $e^{\frac{x^{2}}{2}}$
Method #2
1. Let $u = \frac{x^{2}}{2}$ .
  
  Then let $du = x dx$ and substitute $du$ :
  
  $\int e^{u}\, du$
  1. The integral of the exponential function is itself.
    
    $\int e^{u}\, du = e^{u}$
  Now substitute $u$ back in:
  
  $e^{\frac{x^{2}}{2}}$
Now simplify:

$e^{\frac{x^{2}}{2}}$
Add the constant of integration:

$e^{\frac{x^{2}}{2}}+ \mathrm{constant}$

The answer is:

$e^{\frac{x^{2}}{2}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                  
 |                   
 |     2            2
 |    x            x 
 |    --           --
 |    2            2 
 | x*e   dx = C + e  
 |                   
/

$$\int x e^{\frac{x^{2}}{2}}\, dx = C + e^{\frac{x^{2}}{2}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

      1/2
-1 + e

$$-1 + e^{\frac{1}{2}}$$

      1/2
-1 + e

$$-1 + e^{\frac{1}{2}}$$

-1 + exp(1/2)

Numerical answer [src]

0.648721270700128

0.648721270700128

The graph

Use the examples entering the upper and lower limits of integration.

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How to use it?

Identical expressions

Integral of x*exp(x^2/2) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2