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Solving integrals/ sin(ax+b)

Integral of sin(ax+b) dx

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The solution

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