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Integral of 2^(cosx)sinx dx

The solution

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 90                  
  /                  
 |                   
 |   cos(x)          
 |  2      *sin(x) dx
 |                   
/                    
0

$$\int\limits_{0}^{90} 2^{\cos{\left(x \right)}} \sin{\left(x \right)}\, dx$$

Integral(2^cos(x)*sin(x), (x, 0, 90))

Detail solution

Let $u = \cos{\left(x \right)}$ .

Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :

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

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                               
 |                          cos(x)
 |  cos(x)                 2      
 | 2      *sin(x) dx = C - -------
 |                          log(2)
/

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

Simplify

The graph

Plot the graph f(x)

The answer [src]

          cos(90)
  2      2       
------ - --------
log(2)    log(2)

$$- \frac{1}{2^{- \cos{\left(90 \right)}} \log{\left(2 \right)}} + \frac{2}{\log{\left(2 \right)}}$$

          cos(90)
  2      2       
------ - --------
log(2)    log(2)

$$- \frac{1}{2^{- \cos{\left(90 \right)}} \log{\left(2 \right)}} + \frac{2}{\log{\left(2 \right)}}$$

2/log(2) - 2^cos(90)/log(2)

Numerical answer [src]

1.8278896077875

1.8278896077875

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

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Identical expressions

Integral of 2^(cosx)sinx dx

Limits of integration:

The graph:

Piecewise:

The solution