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Solving integrals/ sinax*ln(cosax)

Integral of sinax*ln(cosax) dx

Limits of integration:

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The graph:

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Piecewise:

The solution

You have entered [src] 
  1                          
  /                          
 |                           
 |  sin(a*x)*log(cos(a*x)) dx
 |                           
/                            
0
$$\int\limits_{0}^{1} \log{\left(\cos{\left(a x \right)} \right)} \sin{\left(a x \right)}\, dx$$ 
Integral(sin(a*x)*log(cos(a*x)), (x, 0, 1))
The answer (Indefinite) [src] 
  /                                //-cos(a*x)*log(cos(a*x)) + cos(a*x)            \
 |                                 ||----------------------------------  for a != 0|
 | sin(a*x)*log(cos(a*x)) dx = C + |<                a                             |
 |                                 ||                                              |
/                                  \\                0                   otherwise /
$$\int \log{\left(\cos{\left(a x \right)} \right)} \sin{\left(a x \right)}\, dx = C + \begin{cases} \frac{- \log{\left(\cos{\left(a x \right)} \right)} \cos{\left(a x \right)} + \cos{\left(a x \right)}}{a} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases}$$
Simplify
The answer [src] 
/  1   cos(a)   cos(a)*log(cos(a))                                  
|- - + ------ - ------------------  for And(a > -oo, a < oo, a != 0)
<  a     a              a                                           
|                                                                   
\                0                             otherwise
$$\begin{cases} - \frac{\log{\left(\cos{\left(a \right)} \right)} \cos{\left(a \right)}}{a} + \frac{\cos{\left(a \right)}}{a} - \frac{1}{a} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\0 & \text{otherwise} \end{cases}$$ 
=
= 
/  1   cos(a)   cos(a)*log(cos(a))                                  
|- - + ------ - ------------------  for And(a > -oo, a < oo, a != 0)
<  a     a              a                                           
|                                                                   
\                0                             otherwise
$$\begin{cases} - \frac{\log{\left(\cos{\left(a \right)} \right)} \cos{\left(a \right)}}{a} + \frac{\cos{\left(a \right)}}{a} - \frac{1}{a} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\0 & \text{otherwise} \end{cases}$$ 
Piecewise((-1/a + cos(a)/a - cos(a)*log(cos(a))/a, (a > -oo)∧(a < oo)∧(Ne(a, 0))), (0, True))

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

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