Mister Exam

Other calculators

Solving integrals/ -4x*exp(4x)

Integral of -4x*exp(4x) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1             
  /             
 |              
 |        4*x   
 |  -4*x*e    dx
 |              
/               
0
∫01−4xe4x dx\int\limits_{0}^{1} - 4 x e^{4 x}\, dx 
Integral((-4*x)*exp(4*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)=−4xu{\left(x \right)} = - 4 x and let dv⁡(x)=e4x\operatorname{dv}{\left(x \right)} = e^{4 x}.

    Then du⁡(x)=−4\operatorname{du}{\left(x \right)} = -4.

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

    1. Let u=4xu = 4 x.

      Then let du=4dxdu = 4 dx and substitute du4\frac{du}{4}:

      ∫eu4 du\int \frac{e^{u}}{4}\, du

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

        False\text{False}

        1. The integral of the exponential function is itself.

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

        So, the result is: eu4\frac{e^{u}}{4}

      Now substitute uu back in:

      e4x4\frac{e^{4 x}}{4}

    Now evaluate the sub-integral.

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

    ∫(−e4x) dx=−∫e4x dx\int \left(- e^{4 x}\right)\, dx = - \int e^{4 x}\, dx

    1. Let u=4xu = 4 x.

      Then let du=4dxdu = 4 dx and substitute du4\frac{du}{4}:

      ∫eu4 du\int \frac{e^{u}}{4}\, du

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

        False\text{False}

        1. The integral of the exponential function is itself.

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

        So, the result is: eu4\frac{e^{u}}{4}

      Now substitute uu back in:

      e4x4\frac{e^{4 x}}{4}

    So, the result is: −e4x4- \frac{e^{4 x}}{4}

  3. Now simplify:

    (14−x)e4x\left(\frac{1}{4} - x\right) e^{4 x}

  4. Add the constant of integration:

    (14−x)e4x+constant\left(\frac{1}{4} - x\right) e^{4 x}+ \mathrm{constant}

The answer is:

(14−x)e4x+constant\left(\frac{1}{4} - x\right) e^{4 x}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                
 |                     4*x         
 |       4*x          e         4*x
 | -4*x*e    dx = C + ---- - x*e   
 |                     4           
/
∫−4xe4x dx=C−xe4x+e4x4\int - 4 x e^{4 x}\, dx = C - x e^{4 x} + \frac{e^{4 x}}{4}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-250250
Plot the graph f(x)
The answer [src] 
         4
  1   3*e 
- - - ----
  4    4
−3e44−14- \frac{3 e^{4}}{4} - \frac{1}{4} 
=
= 
         4
  1   3*e 
- - - ----
  4    4
−3e44−14- \frac{3 e^{4}}{4} - \frac{1}{4} 
-1/4 - 3*exp(4)/4
Numerical answer [src] 
-41.1986125248582
-41.1986125248582

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

    © Mister Exam – Calculator