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Similar expressions
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Solving integrals/ 6^(sin(x))*cos(x)

Integral of 6^(sin(x))*cos(x) dx

The solution

You have entered [src]

 pi                  
 --                  
 2                   
  /                  
 |                   
 |   sin(x)          
 |  6      *cos(x) dx
 |                   
/                    
0

$$\int\limits_{0}^{\frac{\pi}{2}} 6^{\sin{\left(x \right)}} \cos{\left(x \right)}\, dx$$

Integral(6^sin(x)*cos(x), (x, 0, pi/2))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = 6^{\sin{\left(x \right)}}$ .
  
  Then let $du = 6^{\sin{\left(x \right)}} \log{\left(6 \right)} \cos{\left(x \right)} dx$ and substitute $\frac{du}{\log{\left(6 \right)}}$ :
  
  $\int \frac{1}{\log{\left(6 \right)}^{2}}\, du$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{1}{\log{\left(6 \right)}}\, du = \frac{\int 1\, du}{\log{\left(6 \right)}}$
    1. The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
    So, the result is: $\frac{u}{\log{\left(6 \right)}}$
  Now substitute $u$ back in:
  
  $\frac{6^{\sin{\left(x \right)}}}{\log{\left(6 \right)}}$
Method #2
1. Let $u = \sin{\left(x \right)}$ .
  
  Then let $du = \cos{\left(x \right)} dx$ and substitute $du$ :
  
  $\int 6^{u}\, du$
  1. The integral of an exponential function is itself divided by the natural logarithm of the base.
    
    $\int 6^{u}\, du = \frac{6^{u}}{\log{\left(6 \right)}}$
  Now substitute $u$ back in:
  
  $\frac{6^{\sin{\left(x \right)}}}{\log{\left(6 \right)}}$
Add the constant of integration:

$\frac{6^{\sin{\left(x \right)}}}{\log{\left(6 \right)}}+ \mathrm{constant}$

The answer is:

$\frac{6^{\sin{\left(x \right)}}}{\log{\left(6 \right)}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                               
 |                          sin(x)
 |  sin(x)                 6      
 | 6      *cos(x) dx = C + -------
 |                          log(6)
/

$${{6^{\sin x}}\over{\log 6}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

  5   
------
log(6)

$$\frac{5}{\log{\left(6 \right)}}$$

  5   
------
log(6)

$$\frac{5}{\log{\left(6 \right)}}$$

Simplify

Numerical answer [src]

2.79055313275624

2.79055313275624

The graph

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

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Identical expressions

Similar expressions

Integral of 6^(sin(x))*cos(x) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2