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Identical expressions
exp^(2x)+exp^(-3x)
exponent of to the power of (2x) plus exponent of to the power of ( minus 3x)
exp(2x)+exp(-3x)
exp2x+exp-3x
exp^2x+exp^-3x
exp^(2x)+exp^(-3x)dx
Similar expressions
exp^(2x)+exp^(3x)
exp^(2x)-exp^(-3x)

Integral of exp^(2x)+exp^(-3x) dx

The solution

You have entered [src]

  1                  
  /                  
 |                   
 |  / 2*x    -3*x\   
 |  \E    + E    / dx
 |                   
/                    
0

$$\int\limits_{0}^{1} \left(e^{2 x} + e^{- 3 x}\right)\, dx$$

Integral(E^(2*x) + E^(-3*x), (x, 0, 1))

Detail solution

Integrate term-by-term:
1. Let $u = 2 x$ .
  
  Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
  
  $\int \frac{e^{u}}{2}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\text{False}$
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    So, the result is: $\frac{e^{u}}{2}$
  Now substitute $u$ back in:
  
  $\frac{e^{2 x}}{2}$
1. Let $u = - 3 x$ .
  
  Then let $du = - 3 dx$ and substitute $- \frac{du}{3}$ :
  
  $\int \left(- \frac{e^{u}}{3}\right)\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\text{False}$
    1. The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
    So, the result is: $- \frac{e^{u}}{3}$
  Now substitute $u$ back in:
  
  $- \frac{e^{- 3 x}}{3}$
The result is: $\frac{e^{2 x}}{2} - \frac{e^{- 3 x}}{3}$
Now simplify:

$\frac{\left(3 e^{5 x} - 2\right) e^{- 3 x}}{6}$
Add the constant of integration:

$\frac{\left(3 e^{5 x} - 2\right) e^{- 3 x}}{6}+ \mathrm{constant}$

The answer is:

$\frac{\left(3 e^{5 x} - 2\right) e^{- 3 x}}{6}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                    
 |                          2*x    -3*x
 | / 2*x    -3*x\          e      e    
 | \E    + E    / dx = C + ---- - -----
 |                          2       3  
/

$$\int \left(e^{2 x} + e^{- 3 x}\right)\, dx = C + \frac{e^{2 x}}{2} - \frac{e^{- 3 x}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

       2    -3
  1   e    e  
- - + -- - ---
  6   2     3

$$- \frac{1}{6} - \frac{1}{3 e^{3}} + \frac{e^{2}}{2}$$

       2    -3
  1   e    e  
- - + -- - ---
  6   2     3

$$- \frac{1}{6} - \frac{1}{3 e^{3}} + \frac{e^{2}}{2}$$

-1/6 + exp(2)/2 - exp(-3)/3

Numerical answer [src]

3.5112656933427

3.5112656933427

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

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Identical expressions

Similar expressions

Integral of exp^(2x)+exp^(-3x) dx

Limits of integration:

The graph:

Piecewise:

The solution