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3*exp(-2*x)
Solving integrals/ 3*exp(-2*x)

Integral of 3*exp(-2*x) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
  1           
  /           
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 |     -2*x   
 |  3*e     dx
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0
∫013e−2x dx\int\limits_{0}^{1} 3 e^{- 2 x}\, dx 
Integral(3*exp(-2*x), (x, 0, 1))
Detail solution

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

    ∫3e−2x dx=3∫e−2x dx\int 3 e^{- 2 x}\, dx = 3 \int e^{- 2 x}\, dx

    1. Let u=−2xu = - 2 x.

      Then let du=−2dxdu = - 2 dx and substitute −du2- \frac{du}{2}:

      ∫(−eu2) du\int \left(- \frac{e^{u}}{2}\right)\, 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: −eu2- \frac{e^{u}}{2}

      Now substitute uu back in:

      −e−2x2- \frac{e^{- 2 x}}{2}

    So, the result is: −3e−2x2- \frac{3 e^{- 2 x}}{2}

  2. Add the constant of integration:

    −3e−2x2+constant- \frac{3 e^{- 2 x}}{2}+ \mathrm{constant}

The answer is:

−3e−2x2+constant- \frac{3 e^{- 2 x}}{2}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                        
 |                     -2*x
 |    -2*x          3*e    
 | 3*e     dx = C - -------
 |                     2   
/
∫3e−2x dx=C−3e−2x2\int 3 e^{- 2 x}\, dx = C - \frac{3 e^{- 2 x}}{2}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.905-5
Plot the graph f(x)
The answer [src] 
       -2
3   3*e  
- - -----
2     2
32−32e2\frac{3}{2} - \frac{3}{2 e^{2}} 
=
= 
       -2
3   3*e  
- - -----
2     2
32−32e2\frac{3}{2} - \frac{3}{2 e^{2}} 
3/2 - 3*exp(-2)/2
Numerical answer [src] 
1.29699707514508
1.29699707514508
The graph
Integral of 3*exp(-2*x) dx

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

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