Mister Exam

Other calculators

x^2*e^(-x)*dx
Solving integrals/ x^2*e^(-x)*dx

You entered:

x^2*e^(-x)*dx

What you mean?

x^2*dx/e^x
 
x^((2*e)^(-x))*dx
 
x^((2*e)^(-x))*dx

Integral of x^2*e^(-x)*dx dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1            
  /            
 |             
 |   2  -x     
 |  x *e  *1 dx
 |             
/              
0
∫01x2e−x1 dx\int\limits_{0}^{1} x^{2} e^{- x} 1\, dx 
Integral(x^2*1/E^(1*x), (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)=e−x\operatorname{dv}{\left(x \right)} = e^{- x}.

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

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

    1. There are multiple ways to do this integral.

      Method #1

      1. Let u=−xu = - x.

        Then let du=−dxdu = - dx and substitute −du- du:

        ∫eu du\int e^{u}\, du

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

          ∫(−eu) du=−∫eu du\int \left(- e^{u}\right)\, du = - \int e^{u}\, du

          1. The integral of the exponential function is itself.

            ∫eu du=eu\int e^{u}\, du = e^{u}

          So, the result is: −eu- e^{u}

        Now substitute uu back in:

        −e−x- e^{- x}

      Method #2

      1. Let u=e−xu = e^{- x}.

        Then let du=−e−xdxdu = - e^{- x} dx and substitute −du- du:

        ∫1 du\int 1\, du

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

          ∫(−1) du=−∫1 du\int \left(-1\right)\, du = - \int 1\, du

          1. The integral of a constant is the constant times the variable of integration:

            ∫1 du=u\int 1\, du = u

          So, the result is: −u- u

        Now substitute uu back in:

        −e−x- e^{- x}

    Now evaluate the sub-integral.

  2. 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)=−2xu{\left(x \right)} = - 2 x and let dv⁡(x)=e−x\operatorname{dv}{\left(x \right)} = e^{- x}.

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

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

    1. Let u=−xu = - x.

      Then let du=−dxdu = - dx and substitute −du- du:

      ∫eu du\int e^{u}\, du

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

        ∫(−eu) du=−∫eu du\int \left(- e^{u}\right)\, du = - \int e^{u}\, du

        1. The integral of the exponential function is itself.

          ∫eu du=eu\int e^{u}\, du = e^{u}

        So, the result is: −eu- e^{u}

      Now substitute uu back in:

      −e−x- e^{- x}

    Now evaluate the sub-integral.

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

    ∫2e−x dx=2∫e−x dx\int 2 e^{- x}\, dx = 2 \int e^{- x}\, dx

    1. Let u=−xu = - x.

      Then let du=−dxdu = - dx and substitute −du- du:

      ∫eu du\int e^{u}\, du

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

        ∫(−eu) du=−∫eu du\int \left(- e^{u}\right)\, du = - \int e^{u}\, du

        1. The integral of the exponential function is itself.

          ∫eu du=eu\int e^{u}\, du = e^{u}

        So, the result is: −eu- e^{u}

      Now substitute uu back in:

      −e−x- e^{- x}

    So, the result is: −2e−x- 2 e^{- x}

  4. Now simplify:

    (−x2−2x−2)e−x\left(- x^{2} - 2 x - 2\right) e^{- x}

  5. Add the constant of integration:

    (−x2−2x−2)e−x+constant\left(- x^{2} - 2 x - 2\right) e^{- x}+ \mathrm{constant}

The answer is:

(−x2−2x−2)e−x+constant\left(- x^{2} - 2 x - 2\right) e^{- x}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                          
 |                                           
 |  2  -x               -x    2  -x        -x
 | x *e  *1 dx = C - 2*e   - x *e   - 2*x*e  
 |                                           
/
(−x2−2 x−2) e−x\left(-x^2-2\,x-2\right)\,e^ {- x }
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.902.5-2.5
Plot the graph f(x)
The answer [src] 
       -1
2 - 5*e
2−5 e−12-5\,e^ {- 1 } 
=
= 
       -1
2 - 5*e
2−5e2 - \frac{5}{e}
Упростить
Numerical answer [src] 
0.160602794142788
0.160602794142788
The graph
Integral of x^2*e^(-x)*dx dx

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

    © Mister Exam – Calculator