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Integral of 3^(-x) dx

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$$\int\limits_{-2}^{2} 3^{- x}\, dx$$

Integral(3^(-x), (x, -2, 2))

Detail solution

Let $u = - x$ .

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

$\int \left(- 3^{u}\right)\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 3^{u}\, du = - \int 3^{u}\, du$
  1. The integral of an exponential function is itself divided by the natural logarithm of the base.
    
    $\int 3^{u}\, du = \frac{3^{u}}{\log{\left(3 \right)}}$
  So, the result is: $- \frac{3^{u}}{\log{\left(3 \right)}}$
Now substitute $u$ back in:

$- \frac{3^{- x}}{\log{\left(3 \right)}}$
Add the constant of integration:

$- \frac{3^{- x}}{\log{\left(3 \right)}}+ \mathrm{constant}$

The answer is:

$- \frac{3^{- x}}{\log{\left(3 \right)}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                   
 |                -x  
 |  -x           3    
 | 3   dx = C - ------
 |              log(3)
/

$$\int 3^{- x}\, dx = C - \frac{3^{- x}}{\log{\left(3 \right)}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

   80   
--------
9*log(3)

$$\frac{80}{9 \log{\left(3 \right)}}$$

   80   
--------
9*log(3)

$$\frac{80}{9 \log{\left(3 \right)}}$$

80/(9*log(3))

Numerical answer [src]

8.09101534779411

8.09101534779411

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