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Solving integrals/ 5^(2*x)

Integral of 5^(2*x) dx

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$$\int\limits_{0}^{1} 5^{2 x}\, dx$$

Integral(5^(2*x), (x, 0, 1))

Detail solution

Let $u = 2 x$ .

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

$\int \frac{5^{u}}{2}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 5^{u}\, du = \frac{\int 5^{u}\, du}{2}$
  1. The integral of an exponential function is itself divided by the natural logarithm of the base.
    
    $\int 5^{u}\, du = \frac{5^{u}}{\log{\left(5 \right)}}$
  So, the result is: $\frac{5^{u}}{2 \log{\left(5 \right)}}$
Now substitute $u$ back in:

$\frac{5^{2 x}}{2 \log{\left(5 \right)}}$
Now simplify:

$\frac{25^{x}}{2 \log{\left(5 \right)}}$
Add the constant of integration:

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

The answer is:

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

The answer (Indefinite) [src]

  /                      
 |                  2*x  
 |  2*x            5     
 | 5    dx = C + --------
 |               2*log(5)
/

$$\int 5^{2 x}\, dx = \frac{5^{2 x}}{2 \log{\left(5 \right)}} + C$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  12  
------
log(5)

$$\frac{12}{\log{\left(5 \right)}}$$

  12  
------
log(5)

$$\frac{12}{\log{\left(5 \right)}}$$

12/log(5)

Numerical answer [src]

7.45601921471534

7.45601921471534

The graph

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