Integral of 3sinxsinnx dx
The solution
The answer (Indefinite)
[src]
// 2 2 \
|| 3*x*cos (x) 3*x*sin (x) 3*cos(x)*sin(x) |
||- ----------- - ----------- + --------------- for n = -1|
|| 2 2 2 |
|| |
/ || 2 2 |
| || 3*cos(x)*sin(x) 3*x*cos (x) 3*x*sin (x) |
| 3*sin(x)*sin(n*x) dx = C + |<- --------------- + ----------- + ----------- for n = 1 |
| || 2 2 2 |
/ || |
|| 3*cos(x)*sin(n*x) 3*n*cos(n*x)*sin(x) |
|| ----------------- - ------------------- otherwise |
|| 2 2 |
|| -1 + n -1 + n |
\\ /
$$\int 3 \sin{\left(x \right)} \sin{\left(n x \right)}\, dx = C + \begin{cases} - \frac{3 x \sin^{2}{\left(x \right)}}{2} - \frac{3 x \cos^{2}{\left(x \right)}}{2} + \frac{3 \sin{\left(x \right)} \cos{\left(x \right)}}{2} & \text{for}\: n = -1 \\\frac{3 x \sin^{2}{\left(x \right)}}{2} + \frac{3 x \cos^{2}{\left(x \right)}}{2} - \frac{3 \sin{\left(x \right)} \cos{\left(x \right)}}{2} & \text{for}\: n = 1 \\- \frac{3 n \sin{\left(x \right)} \cos{\left(n x \right)}}{n^{2} - 1} + \frac{3 \sin{\left(n x \right)} \cos{\left(x \right)}}{n^{2} - 1} & \text{otherwise} \end{cases}$$
/ 2 2
| 3*cos (1) 3*sin (1) 3*cos(1)*sin(1)
|- --------- - --------- + --------------- for n = -1
| 2 2 2
|
| 2 2
| 3*cos (1) 3*sin (1) 3*cos(1)*sin(1)
< --------- + --------- - --------------- for n = 1
| 2 2 2
|
| 3*cos(1)*sin(n) 3*n*cos(n)*sin(1)
| --------------- - ----------------- otherwise
| 2 2
| -1 + n -1 + n
\
$$\begin{cases} - \frac{3 \sin^{2}{\left(1 \right)}}{2} - \frac{3 \cos^{2}{\left(1 \right)}}{2} + \frac{3 \sin{\left(1 \right)} \cos{\left(1 \right)}}{2} & \text{for}\: n = -1 \\- \frac{3 \sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{3 \cos^{2}{\left(1 \right)}}{2} + \frac{3 \sin^{2}{\left(1 \right)}}{2} & \text{for}\: n = 1 \\- \frac{3 n \sin{\left(1 \right)} \cos{\left(n \right)}}{n^{2} - 1} + \frac{3 \sin{\left(n \right)} \cos{\left(1 \right)}}{n^{2} - 1} & \text{otherwise} \end{cases}$$
=
/ 2 2
| 3*cos (1) 3*sin (1) 3*cos(1)*sin(1)
|- --------- - --------- + --------------- for n = -1
| 2 2 2
|
| 2 2
| 3*cos (1) 3*sin (1) 3*cos(1)*sin(1)
< --------- + --------- - --------------- for n = 1
| 2 2 2
|
| 3*cos(1)*sin(n) 3*n*cos(n)*sin(1)
| --------------- - ----------------- otherwise
| 2 2
| -1 + n -1 + n
\
$$\begin{cases} - \frac{3 \sin^{2}{\left(1 \right)}}{2} - \frac{3 \cos^{2}{\left(1 \right)}}{2} + \frac{3 \sin{\left(1 \right)} \cos{\left(1 \right)}}{2} & \text{for}\: n = -1 \\- \frac{3 \sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{3 \cos^{2}{\left(1 \right)}}{2} + \frac{3 \sin^{2}{\left(1 \right)}}{2} & \text{for}\: n = 1 \\- \frac{3 n \sin{\left(1 \right)} \cos{\left(n \right)}}{n^{2} - 1} + \frac{3 \sin{\left(n \right)} \cos{\left(1 \right)}}{n^{2} - 1} & \text{otherwise} \end{cases}$$
Piecewise((-3*cos(1)^2/2 - 3*sin(1)^2/2 + 3*cos(1)*sin(1)/2, n = -1), (3*cos(1)^2/2 + 3*sin(1)^2/2 - 3*cos(1)*sin(1)/2, n = 1), (3*cos(1)*sin(n)/(-1 + n^2) - 3*n*cos(n)*sin(1)/(-1 + n^2), True))
Use the examples entering the upper and lower limits of integration.