Mister Exam

Other calculators


x^3*e^x

You entered:

x^3*e^x

What you mean?

Integral of x^3*e^x dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1         
  /         
 |          
 |   3  x   
 |  x *e  dx
 |          
/           
0           
01x3exdx\int\limits_{0}^{1} x^{3} e^{x}\, dx
Integral(x^3*E^x, (x, 0, 1))
Detail solution
  1. Use integration by parts:

    udv=uvvdu\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

    Let u(x)=x3u{\left(x \right)} = x^{3} and let dv(x)=ex\operatorname{dv}{\left(x \right)} = e^{x}.

    Then du(x)=3x2\operatorname{du}{\left(x \right)} = 3 x^{2}.

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

    1. The integral of the exponential function is itself.

      exdx=ex\int e^{x}\, dx = e^{x}

    Now evaluate the sub-integral.

  2. Use integration by parts:

    udv=uvvdu\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

    Let u(x)=3x2u{\left(x \right)} = 3 x^{2} and let dv(x)=ex\operatorname{dv}{\left(x \right)} = e^{x}.

    Then du(x)=6x\operatorname{du}{\left(x \right)} = 6 x.

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

    1. The integral of the exponential function is itself.

      exdx=ex\int e^{x}\, dx = e^{x}

    Now evaluate the sub-integral.

  3. Use integration by parts:

    udv=uvvdu\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}

    Let u(x)=6xu{\left(x \right)} = 6 x and let dv(x)=ex\operatorname{dv}{\left(x \right)} = e^{x}.

    Then du(x)=6\operatorname{du}{\left(x \right)} = 6.

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

    1. The integral of the exponential function is itself.

      exdx=ex\int e^{x}\, dx = e^{x}

    Now evaluate the sub-integral.

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

    6exdx=6exdx\int 6 e^{x}\, dx = 6 \int e^{x}\, dx

    1. The integral of the exponential function is itself.

      exdx=ex\int e^{x}\, dx = e^{x}

    So, the result is: 6ex6 e^{x}

  5. Now simplify:

    (x33x2+6x6)ex\left(x^{3} - 3 x^{2} + 6 x - 6\right) e^{x}

  6. Add the constant of integration:

    (x33x2+6x6)ex+constant\left(x^{3} - 3 x^{2} + 6 x - 6\right) e^{x}+ \mathrm{constant}


The answer is:

(x33x2+6x6)ex+constant\left(x^{3} - 3 x^{2} + 6 x - 6\right) e^{x}+ \mathrm{constant}

The answer (Indefinite) [src]
  /                                              
 |                                               
 |  3  x             x    3  x      2  x        x
 | x *e  dx = C - 6*e  + x *e  - 3*x *e  + 6*x*e 
 |                                               
/                                                
(x33x2+6x6)ex\left(x^3-3\,x^2+6\,x-6\right)\,e^{x}
The graph
0.001.000.100.200.300.400.500.600.700.800.90-1010
The answer [src]
6 - 2*e
62e6-2\,e
=
=
6 - 2*e
62e6 - 2 e
Numerical answer [src]
0.563436343081909
0.563436343081909
The graph
Integral of x^3*e^x dx

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