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Integral of 5^(-x) dx

You have entered [src]

$$\int\limits_{0}^{\pi} 5^{- x}\, dx$$

Integral(5^(-x), (x, 0, pi))

Detail solution

Let $u = - x$ .

Then let $du = - dx$ and substitute $- du$ :

$\int \left(- 5^{u}\right)\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 5^{u}\, du = - \int 5^{u}\, du$
  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: $- \frac{5^{u}}{\log{\left(5 \right)}}$
Now substitute $u$ back in:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                   
 |                -x  
 |  -x           5    
 | 5   dx = C - ------
 |              log(5)
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

           -pi 
  1       5    
------ - ------
log(5)   log(5)

$$- \frac{1}{5^{\pi} \log{\left(5 \right)}} + \frac{1}{\log{\left(5 \right)}}$$

           -pi 
  1       5    
------ - ------
log(5)   log(5)

$$- \frac{1}{5^{\pi} \log{\left(5 \right)}} + \frac{1}{\log{\left(5 \right)}}$$

1/log(5) - 5^(-pi)/log(5)

Numerical answer [src]

0.617377199284111

0.617377199284111

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