Mister Exam

Other calculators

Solving integrals/ e^(2x)/(1-3e^(2x))

Integral of e^(2x)/(1-3e^(2x)) dx

Limits of integration:

from to
v

The graph:

from to

Piecewise:

The solution

You have entered [src] 
  1              
  /              
 |               
 |      2*x      
 |     E         
 |  ---------- dx
 |         2*x   
 |  1 - 3*E      
 |               
/                
0
∫01e2x1−3e2x dx\int\limits_{0}^{1} \frac{e^{2 x}}{1 - 3 e^{2 x}}\, dx 
Integral(E^(2*x)/(1 - 3*exp(2*x)), (x, 0, 1))
Detail solution

  1. There are multiple ways to do this integral.

    Method #1

    1. Let u=e2xu = e^{2 x}.

      Then let du=2e2xdxdu = 2 e^{2 x} dx and substitute −du- du:

      ∫(−16u−2) du\int \left(- \frac{1}{6 u - 2}\right)\, du

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

        ∫16u−2 du=−∫16u−2 du\int \frac{1}{6 u - 2}\, du = - \int \frac{1}{6 u - 2}\, du

        1. There are multiple ways to do this integral.

          Method #1

          1. Let u=6u−2u = 6 u - 2.

            Then let du=6dudu = 6 du and substitute du6\frac{du}{6}:

            ∫16u du\int \frac{1}{6 u}\, du

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

              ∫1u du=∫1u du6\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{6}

              1. The integral of 1u\frac{1}{u} is log⁡(u)\log{\left(u \right)}.

              So, the result is: log⁡(u)6\frac{\log{\left(u \right)}}{6}

            Now substitute uu back in:

            log⁡(6u−2)6\frac{\log{\left(6 u - 2 \right)}}{6}

          Method #2

          1. Rewrite the integrand:

            16u−2=12(3u−1)\frac{1}{6 u - 2} = \frac{1}{2 \left(3 u - 1\right)}

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

            ∫12(3u−1) du=∫13u−1 du2\int \frac{1}{2 \left(3 u - 1\right)}\, du = \frac{\int \frac{1}{3 u - 1}\, du}{2}

            1. Let u=3u−1u = 3 u - 1.

              Then let du=3dudu = 3 du and substitute du3\frac{du}{3}:

              ∫13u du\int \frac{1}{3 u}\, du

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

                ∫1u du=∫1u du3\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{3}

                1. The integral of 1u\frac{1}{u} is log⁡(u)\log{\left(u \right)}.

                So, the result is: log⁡(u)3\frac{\log{\left(u \right)}}{3}

              Now substitute uu back in:

              log⁡(3u−1)3\frac{\log{\left(3 u - 1 \right)}}{3}

            So, the result is: log⁡(3u−1)6\frac{\log{\left(3 u - 1 \right)}}{6}

        So, the result is: −log⁡(6u−2)6- \frac{\log{\left(6 u - 2 \right)}}{6}

      Now substitute uu back in:

      −log⁡(6e2x−2)6- \frac{\log{\left(6 e^{2 x} - 2 \right)}}{6}

    Method #2

    1. Rewrite the integrand:

      e2x1−3e2x=−e2x3e2x−1\frac{e^{2 x}}{1 - 3 e^{2 x}} = - \frac{e^{2 x}}{3 e^{2 x} - 1}

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

      ∫(−e2x3e2x−1) dx=−∫e2x3e2x−1 dx\int \left(- \frac{e^{2 x}}{3 e^{2 x} - 1}\right)\, dx = - \int \frac{e^{2 x}}{3 e^{2 x} - 1}\, dx

      1. Let u=3e2x−1u = 3 e^{2 x} - 1.

        Then let du=6e2xdxdu = 6 e^{2 x} dx and substitute du6\frac{du}{6}:

        ∫16u du\int \frac{1}{6 u}\, du

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

          ∫1u du=∫1u du6\int \frac{1}{u}\, du = \frac{\int \frac{1}{u}\, du}{6}

          1. The integral of 1u\frac{1}{u} is log⁡(u)\log{\left(u \right)}.

          So, the result is: log⁡(u)6\frac{\log{\left(u \right)}}{6}

        Now substitute uu back in:

        log⁡(3e2x−1)6\frac{\log{\left(3 e^{2 x} - 1 \right)}}{6}

      So, the result is: −log⁡(3e2x−1)6- \frac{\log{\left(3 e^{2 x} - 1 \right)}}{6}

  2. Add the constant of integration:

    −log⁡(6e2x−2)6+constant- \frac{\log{\left(6 e^{2 x} - 2 \right)}}{6}+ \mathrm{constant}

The answer is:

−log⁡(6e2x−2)6+constant- \frac{\log{\left(6 e^{2 x} - 2 \right)}}{6}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                    
 |                                     
 |     2*x                /        2*x\
 |    E                log\-2 + 6*e   /
 | ---------- dx = C - ----------------
 |        2*x                 6        
 | 1 - 3*E                             
 |                                     
/
∫e2x1−3e2x dx=C−log⁡(6e2x−2)6\int \frac{e^{2 x}}{1 - 3 e^{2 x}}\, dx = C - \frac{\log{\left(6 e^{2 x} - 2 \right)}}{6}
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90-1.00.5
Plot the graph f(x)
The answer [src] 
     /  1    2\           
  log|- - + e |           
     \  3     /   log(2/3)
- ------------- + --------
        6            6
−log⁡(−13+e2)6+log⁡(23)6- \frac{\log{\left(- \frac{1}{3} + e^{2} \right)}}{6} + \frac{\log{\left(\frac{2}{3} \right)}}{6} 
=
= 
     /  1    2\           
  log|- - + e |           
     \  3     /   log(2/3)
- ------------- + --------
        6            6
−log⁡(−13+e2)6+log⁡(23)6- \frac{\log{\left(- \frac{1}{3} + e^{2} \right)}}{6} + \frac{\log{\left(\frac{2}{3} \right)}}{6} 
-log(-1/3 + exp(2))/6 + log(2/3)/6
Numerical answer [src] 
-0.393217355908468
-0.393217355908468

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

    © Mister Exam – Calculator