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exp((four *x)/ three)-(two /exp(-x))
exponent of ((4 multiply by x) divide by 3) minus (2 divide by exponent of ( minus x))
exponent of ((four multiply by x) divide by three) minus (two divide by exponent of ( minus x))
exp((4x)/3)-(2/exp(-x))
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exp((4*x) divide by 3)-(2 divide by exp(-x))
exp((4*x)/3)-(2/exp(-x))dx
Similar expressions
exp((4*x)/3)+(2/exp(-x))
exp((4*x)/3)-(2/exp(x))

Solving integrals/ exp((4*x)/3)-(2/exp(-x))

Integral of exp((4*x)/3)-(2/exp(-x)) dx

The solution

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  1                
  /                
 |                 
 |  / 4*x      \   
 |  | ---      |   
 |  |  3     2 |   
 |  |e    - ---| dx
 |  |        -x|   
 |  \       e  /   
 |                 
/                  
0

$$\int\limits_{0}^{1} \left(e^{\frac{4 x}{3}} - \frac{2}{e^{- x}}\right)\, dx$$

Integral(exp((4*x)/3) - 2*exp(x), (x, 0, 1))

Detail solution

Integrate term-by-term:
1. Let $u = \frac{4 x}{3}$ .
  
  Then let $du = \frac{4 dx}{3}$ and substitute $\frac{3 du}{4}$ :
  
  $\int \frac{3 e^{u}}{4}\, du$
  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: $\frac{3 e^{u}}{4}$
  Now substitute $u$ back in:
  
  $\frac{3 e^{\frac{4 x}{3}}}{4}$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \left(- \frac{2}{e^{- x}}\right)\, dx = - 2 \int \frac{1}{e^{- x}}\, dx$
  1. Let $u = e^{- x}$ .
    
    Then let $du = - e^{- x} dx$ and substitute $- du$ :
    
    $\int \left(- \frac{1}{u^{2}}\right)\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \frac{1}{u^{2}}\, du = - \int \frac{1}{u^{2}}\, du$
      1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
        
        $\int \frac{1}{u^{2}}\, du = - \frac{1}{u}$
      So, the result is: $\frac{1}{u}$
    Now substitute $u$ back in:
    
    $e^{x}$
  So, the result is: $- 2 e^{x}$
The result is: $- 2 e^{x} + \frac{3 e^{\frac{4 x}{3}}}{4}$
Now simplify:

$\frac{3 e^{\frac{4 x}{3}}}{4} - 2 e^{x}$
Add the constant of integration:

$\frac{3 e^{\frac{4 x}{3}}}{4} - 2 e^{x}+ \mathrm{constant}$

The answer is:

$\frac{3 e^{\frac{4 x}{3}}}{4} - 2 e^{x}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                   
 |                                 4*x
 | / 4*x      \                    ---
 | | ---      |                     3 
 | |  3     2 |             x   3*e   
 | |e    - ---| dx = C - 2*e  + ------
 | |        -x|                   4   
 | \       e  /                       
 |                                    
/

$$\int \left(e^{\frac{4 x}{3}} - \frac{2}{e^{- x}}\right)\, dx = C - 2 e^{x} + \frac{3 e^{\frac{4 x}{3}}}{4}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

             4/3
5         3*e   
- - 2*E + ------
4           4

$$- 2 e + \frac{5}{4} + \frac{3 e^{\frac{4}{3}}}{4}$$

             4/3
5         3*e   
- - 2*E + ------
4           4

$$- 2 e + \frac{5}{4} + \frac{3 e^{\frac{4}{3}}}{4}$$

5/4 - 2*E + 3*exp(4/3)/4

Numerical answer [src]

-1.34131273590571

-1.34131273590571

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

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Identical expressions

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Integral of exp((4*x)/3)-(2/exp(-x)) dx

Limits of integration:

The graph:

Piecewise:

The solution