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5^(2*x)
Solving integrals/ 5^(2*x)

Integral of 5^(2*x) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
  1        
  /        
 |         
 |   2*x   
 |  5    dx
 |         
/          
0
∫0152x dx\int\limits_{0}^{1} 5^{2 x}\, dx 
Integral(5^(2*x), (x, 0, 1))
Detail solution

  1. Let u=2xu = 2 x.

    Then let du=2dxdu = 2 dx and substitute du2\frac{du}{2}:

    ∫5u2 du\int \frac{5^{u}}{2}\, du

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

      ∫5u du=∫5u du2\int 5^{u}\, du = \frac{\int 5^{u}\, du}{2}

      1. The integral of an exponential function is itself divided by the natural logarithm of the base.

        ∫5u du=5ulog⁡(5)\int 5^{u}\, du = \frac{5^{u}}{\log{\left(5 \right)}}

      So, the result is: 5u2log⁡(5)\frac{5^{u}}{2 \log{\left(5 \right)}}

    Now substitute uu back in:

    52x2log⁡(5)\frac{5^{2 x}}{2 \log{\left(5 \right)}}

  2. Now simplify:

    25x2log⁡(5)\frac{25^{x}}{2 \log{\left(5 \right)}}

  3. Add the constant of integration:

    25x2log⁡(5)+constant\frac{25^{x}}{2 \log{\left(5 \right)}}+ \mathrm{constant}

The answer is:

25x2log⁡(5)+constant\frac{25^{x}}{2 \log{\left(5 \right)}}+ \mathrm{constant}

The answer (Indefinite) [src] 
  /                      
 |                  2*x  
 |  2*x            5     
 | 5    dx = C + --------
 |               2*log(5)
/
∫52x dx=52x2log⁡(5)+C\int 5^{2 x}\, dx = \frac{5^{2 x}}{2 \log{\left(5 \right)}} + C
Simplify
The graph
0.001.000.100.200.300.400.500.600.700.800.90050
Plot the graph f(x)
The answer [src] 
  12  
------
log(5)
12log⁡(5)\frac{12}{\log{\left(5 \right)}} 
=
= 
  12  
------
log(5)
12log⁡(5)\frac{12}{\log{\left(5 \right)}} 
12/log(5)
Numerical answer [src] 
7.45601921471534
7.45601921471534
The graph
Integral of 5^(2*x) dx

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

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