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Solving integrals/ 3^(2*x)

Integral of 3^(2*x) dx

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$$\int\limits_{0}^{1} 3^{2 x}\, dx$$

Integral(3^(2*x), (x, 0, 1))

Detail solution

Let $u = 2 x$ .

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

$\int \frac{3^{u}}{2}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 3^{u}\, du = \frac{\int 3^{u}\, du}{2}$
  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: $\frac{3^{u}}{2 \log{\left(3 \right)}}$
Now substitute $u$ back in:

$\frac{3^{2 x}}{2 \log{\left(3 \right)}}$
Now simplify:

$\frac{9^{x}}{2 \log{\left(3 \right)}}$
Add the constant of integration:

$\frac{9^{x}}{2 \log{\left(3 \right)}}+ \mathrm{constant}$

The answer is:

$\frac{9^{x}}{2 \log{\left(3 \right)}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                      
 |                  2*x  
 |  2*x            3     
 | 3    dx = C + --------
 |               2*log(3)
/

$$\int 3^{2 x}\, dx = \frac{3^{2 x}}{2 \log{\left(3 \right)}} + C$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  4   
------
log(3)

$$\frac{4}{\log{\left(3 \right)}}$$

  4   
------
log(3)

$$\frac{4}{\log{\left(3 \right)}}$$

4/log(3)

Numerical answer [src]

3.64095690650735

3.64095690650735

The graph

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