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Integral of 4^cosx*sinxdx dx

The solution

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  1                  
  /                  
 |                   
 |   cos(x)          
 |  4      *sin(x) dx
 |                   
/                    
0

$$\int\limits_{0}^{1} 4^{\cos{\left(x \right)}} \sin{\left(x \right)}\, dx$$

Integral(4^cos(x)*sin(x), (x, 0, 1))

Detail solution

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

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

$\int \left(- 4^{u}\right)\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 4^{u}\, du = - \int 4^{u}\, du$
  1. The integral of an exponential function is itself divided by the natural logarithm of the base.
    
    $\int 4^{u}\, du = \frac{4^{u}}{\log{\left(4 \right)}}$
  So, the result is: $- \frac{4^{u}}{\log{\left(4 \right)}}$
Now substitute $u$ back in:

$- \frac{4^{\cos{\left(x \right)}}}{\log{\left(4 \right)}}$
Add the constant of integration:

$- \frac{4^{\cos{\left(x \right)}}}{\log{\left(4 \right)}}+ \mathrm{constant}$

The answer is:

$- \frac{4^{\cos{\left(x \right)}}}{\log{\left(4 \right)}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                               
 |                          cos(x)
 |  cos(x)                 4      
 | 4      *sin(x) dx = C - -------
 |                          log(4)
/

$$\int 4^{\cos{\left(x \right)}} \sin{\left(x \right)}\, dx = - \frac{4^{\cos{\left(x \right)}}}{\log{\left(4 \right)}} + C$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

          cos(1) 
  2      4       
------ - --------
log(2)   2*log(2)

$$- \frac{4^{\cos{\left(1 \right)}}}{2 \log{\left(2 \right)}} + \frac{2}{\log{\left(2 \right)}}$$

          cos(1) 
  2      4       
------ - --------
log(2)   2*log(2)

$$- \frac{4^{\cos{\left(1 \right)}}}{2 \log{\left(2 \right)}} + \frac{2}{\log{\left(2 \right)}}$$

2/log(2) - 4^cos(1)/(2*log(2))

Numerical answer [src]

1.35979617708281

1.35979617708281

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

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Identical expressions

Integral of 4^cosx*sinxdx dx

Limits of integration:

The graph:

Piecewise:

The solution