Mister Exam

Other calculators

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

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1            
  /            
 |             
 |      / 2\   
 |   2  \x /   
 |  x *e     dx
 |             
/              
0
∫01x2ex2 dx\int\limits_{0}^{1} x^{2} e^{x^{2}}\, dx 
Integral(x^2*exp(x^2), (x, 0, 1))
Detail solution

  1. Use integration by parts:

    ∫u⁡dv⁡=u⁡v⁡−∫v⁡du⁡\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

    Let u(x)=x2u{\left(x \right)} = x^{2} and let dv⁡(x)=ex2\operatorname{dv}{\left(x \right)} = e^{x^{2}}.

    Then du⁡(x)=2x\operatorname{du}{\left(x \right)} = 2 x.

    To find v(x)v{\left(x \right)}:

      ErfRule(a=1, b=0, c=0, context=exp(x**2), symbol=x)

    Now evaluate the sub-integral.

  2. The integral of a constant times a function is the constant times the integral of the function:

    ∫πxerfi⁡(x) dx=π∫xerfi⁡(x) dx\int \sqrt{\pi} x \operatorname{erfi}{\left(x \right)}\, dx = \sqrt{\pi} \int x \operatorname{erfi}{\left(x \right)}\, dx

    1. Don't know the steps in finding this integral.

      But the integral is

      x2erfi⁡(x)2−xex22π+erfi⁡(x)4\frac{x^{2} \operatorname{erfi}{\left(x \right)}}{2} - \frac{x e^{x^{2}}}{2 \sqrt{\pi}} + \frac{\operatorname{erfi}{\left(x \right)}}{4}

    So, the result is: π(x2erfi⁡(x)2−xex22π+erfi⁡(x)4)\sqrt{\pi} \left(\frac{x^{2} \operatorname{erfi}{\left(x \right)}}{2} - \frac{x e^{x^{2}}}{2 \sqrt{\pi}} + \frac{\operatorname{erfi}{\left(x \right)}}{4}\right)

  3. Now simplify:

    xex22−πerfi⁡(x)4\frac{x e^{x^{2}}}{2} - \frac{\sqrt{\pi} \operatorname{erfi}{\left(x \right)}}{4}

  4. Add the constant of integration:

    xex22−πerfi⁡(x)4+constant\frac{x e^{x^{2}}}{2} - \frac{\sqrt{\pi} \operatorname{erfi}{\left(x \right)}}{4}+ \mathrm{constant}

The answer is:

xex22−πerfi⁡(x)4+constant\frac{x e^{x^{2}}}{2} - \frac{\sqrt{\pi} \operatorname{erfi}{\left(x \right)}}{4}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                                                              
 |                          /                          / 2\ \                    
 |     / 2\                 |           2              \x / |     ____  2        
 |  2  \x /            ____ |erfi(x)   x *erfi(x)   x*e     |   \/ pi *x *erfi(x)
 | x *e     dx = C - \/ pi *|------- + ---------- - --------| + -----------------
 |                          |   4          2            ____|           2        
/                           \                       2*\/ pi /
∫x2ex2 dx=C+πx2erfi⁡(x)2−π(x2erfi⁡(x)2−xex22π+erfi⁡(x)4)\int x^{2} e^{x^{2}}\, dx = C + \frac{\sqrt{\pi} x^{2} \operatorname{erfi}{\left(x \right)}}{2} - \sqrt{\pi} \left(\frac{x^{2} \operatorname{erfi}{\left(x \right)}}{2} - \frac{x e^{x^{2}}}{2 \sqrt{\pi}} + \frac{\operatorname{erfi}{\left(x \right)}}{4}\right)
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.900.05.0
Plot the graph f(x)
The answer [src] 
      ____        
E   \/ pi *erfi(1)
- - --------------
2         4
−πerfi⁡(1)4+e2- \frac{\sqrt{\pi} \operatorname{erfi}{\left(1 \right)}}{4} + \frac{e}{2} 
=
= 
      ____        
E   \/ pi *erfi(1)
- - --------------
2         4
−πerfi⁡(1)4+e2- \frac{\sqrt{\pi} \operatorname{erfi}{\left(1 \right)}}{4} + \frac{e}{2} 
E/2 - sqrt(pi)*erfi(1)/4
Numerical answer [src] 
0.627815041275932
0.627815041275932
The graph
Integral of x^2*exp(x^2) dx

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

    © Mister Exam – Calculator