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

Integral of x*2^(3x-1) dx

Limits of integration:

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Piecewise:

The solution

You have entered [src] 
  1              
  /              
 |               
 |     3*x - 1   
 |  x*2        dx
 |               
/                
0
∫0123x−1x dx\int\limits_{0}^{1} 2^{3 x - 1} x\, dx 
Integral(x*2^(3*x - 1*1), (x, 0, 1))
Detail solution

  1. Rewrite the integrand:

    23x−1x=23xx22^{3 x - 1} x = \frac{2^{3 x} x}{2}

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

    ∫23xx2 dx=∫23xx dx2\int \frac{2^{3 x} x}{2}\, dx = \frac{\int 2^{3 x} x\, dx}{2}

    1. Don't know the steps in finding this integral.

      But the integral is

      23x(3xlog⁡(2)−1)9log⁡(2)2\frac{2^{3 x} \left(3 x \log{\left(2 \right)} - 1\right)}{9 \log{\left(2 \right)}^{2}}

    So, the result is: 23x(3xlog⁡(2)−1)18log⁡(2)2\frac{2^{3 x} \left(3 x \log{\left(2 \right)} - 1\right)}{18 \log{\left(2 \right)}^{2}}

  3. Now simplify:

    23x(xlog⁡(8)−1)18log⁡(2)2\frac{2^{3 x} \left(x \log{\left(8 \right)} - 1\right)}{18 \log{\left(2 \right)}^{2}}

  4. Add the constant of integration:

    23x(xlog⁡(8)−1)18log⁡(2)2+constant\frac{2^{3 x} \left(x \log{\left(8 \right)} - 1\right)}{18 \log{\left(2 \right)}^{2}}+ \mathrm{constant}

The answer is:

23x(xlog⁡(8)−1)18log⁡(2)2+constant\frac{2^{3 x} \left(x \log{\left(8 \right)} - 1\right)}{18 \log{\left(2 \right)}^{2}}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                                          
 |                      3*x                  
 |    3*x - 1          2   *(-1 + 3*x*log(2))
 | x*2        dx = C + ----------------------
 |                                 2         
/                            18*log (2)
(3 log⁡2 x−1) e3 log⁡2 x18 (log⁡2)2{{\left(3\,\log 2\,x-1\right)\,e^{3\,\log 2\,x}}\over{18\,\left( \log 2\right)^2}}
Simplify
The graph
02468-8-6-4-2-1010-500000000010000000000
Plot the graph f(x)
The answer [src] 
    1        4*(-1 + 3*log(2))
---------- + -----------------
      2               2       
18*log (2)       9*log (2)
12 log⁡2−49 (log⁡2)2+118 (log⁡2)2{{12\,\log 2-4}\over{9\,\left(\log 2\right)^2}}+{{1}\over{18\, \left(\log 2\right)^2}} 
=
= 
    1        4*(-1 + 3*log(2))
---------- + -----------------
      2               2       
18*log (2)       9*log (2)
118log⁡(2)2+4(−1+3log⁡(2))9log⁡(2)2\frac{1}{18 \log{\left(2 \right)}^{2}} + \frac{4 \left(-1 + 3 \log{\left(2 \right)}\right)}{9 \log{\left(2 \right)}^{2}}
Simplify
Numerical answer [src] 
1.11417211746088
1.11417211746088
The graph
Integral of x*2^(3x-1) dx

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

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