Integral of dx/e^(9x) dx

The solution

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$$\int\limits_{0}^{1} \frac{1}{e^{9 x}}\, dx$$

Integral(1/(E^(9*x)), (x, 0, 1))

Detail solution

Let $u = e^{9 x}$ .

Then let $du = 9 e^{9 x} dx$ and substitute $\frac{du}{9}$ :

$\int \frac{1}{9 u^{2}}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \frac{1}{u^{2}}\, du = \frac{\int \frac{1}{u^{2}}\, du}{9}$
  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int \frac{1}{u^{2}}\, du = - \frac{1}{u}$
  So, the result is: $- \frac{1}{9 u}$
Now substitute $u$ back in:

$- \frac{e^{- 9 x}}{9}$
Add the constant of integration:

$- \frac{e^{- 9 x}}{9}+ \mathrm{constant}$

The answer is:

$- \frac{e^{- 9 x}}{9}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                   
 |                -9*x
 |  1            e    
 | ---- dx = C - -----
 |  9*x            9  
 | E                  
 |                    
/

$$\int \frac{1}{e^{9 x}}\, dx = C - \frac{e^{- 9 x}}{9}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

     -9
1   e  
- - ---
9    9

$$\frac{1}{9} - \frac{1}{9 e^{9}}$$

     -9
1   e  
- - ---
9    9

$$\frac{1}{9} - \frac{1}{9 e^{9}}$$

1/9 - exp(-9)/9

Numerical answer [src]

0.111097398910657

0.111097398910657

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

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Identical expressions

Integral of dx/e^(9x) dx

Limits of integration:

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Piecewise:

The solution