Mister Exam

Other calculators

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

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src]
  1                
  /                
 |                 
 |  / 4*x      \   
 |  | ---      |   
 |  |  3     2 |   
 |  |e    - ---| dx
 |  |        -x|   
 |  \       e  /   
 |                 
/                  
0                  
01(e4x32ex)dx\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
  1. Integrate term-by-term:

    1. Let u=4x3u = \frac{4 x}{3}.

      Then let du=4dx3du = \frac{4 dx}{3} and substitute 3du4\frac{3 du}{4}:

      3eu4du\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:

        False\text{False}

        1. The integral of the exponential function is itself.

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

        So, the result is: 3eu4\frac{3 e^{u}}{4}

      Now substitute uu back in:

      3e4x34\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:

      (2ex)dx=21exdx\int \left(- \frac{2}{e^{- x}}\right)\, dx = - 2 \int \frac{1}{e^{- x}}\, dx

      1. Let u=exu = e^{- x}.

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

        (1u2)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:

          1u2du=1u2du\int \frac{1}{u^{2}}\, du = - \int \frac{1}{u^{2}}\, du

          1. The integral of unu^{n} is un+1n+1\frac{u^{n + 1}}{n + 1} when n1n \neq -1:

            1u2du=1u\int \frac{1}{u^{2}}\, du = - \frac{1}{u}

          So, the result is: 1u\frac{1}{u}

        Now substitute uu back in:

        exe^{x}

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

    The result is: 2ex+3e4x34- 2 e^{x} + \frac{3 e^{\frac{4 x}{3}}}{4}

  2. Now simplify:

    3e4x342ex\frac{3 e^{\frac{4 x}{3}}}{4} - 2 e^{x}

  3. Add the constant of integration:

    3e4x342ex+constant\frac{3 e^{\frac{4 x}{3}}}{4} - 2 e^{x}+ \mathrm{constant}


The answer is:

3e4x342ex+constant\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  /                       
 |                                    
/                                     
(e4x32ex)dx=C2ex+3e4x34\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}
The graph
0.001.000.100.200.300.400.500.600.700.800.900-4
The answer [src]
             4/3
5         3*e   
- - 2*E + ------
4           4   
2e+54+3e434- 2 e + \frac{5}{4} + \frac{3 e^{\frac{4}{3}}}{4}
=
=
             4/3
5         3*e   
- - 2*E + ------
4           4   
2e+54+3e434- 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.