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Identical expressions
- 3sinxsinnx
- 3 sinus of x sinus of nx
- 3sinxsinnxdx
Integral of 3sinxsinnx dx
The solution
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[src]
1 / | | 3*sin(x)*sin(n*x) dx | / 0
$$\int\limits_{0}^{1} 3 \sin{\left(x \right)} \sin{\left(n x \right)}\, dx$$
Integral((3*sin(x))*sin(n*x), (x, 0, 1))
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 |
\\ /
$$\int 3 \sin{\left(x \right)} \sin{\left(n x \right)}\, dx = C + \begin{cases} - \frac{3 x \sin^{2}{\left(x \right)}}{2} - \frac{3 x \cos^{2}{\left(x \right)}}{2} + \frac{3 \sin{\left(x \right)} \cos{\left(x \right)}}{2} & \text{for}\: n = -1 \\\frac{3 x \sin^{2}{\left(x \right)}}{2} + \frac{3 x \cos^{2}{\left(x \right)}}{2} - \frac{3 \sin{\left(x \right)} \cos{\left(x \right)}}{2} & \text{for}\: n = 1 \\- \frac{3 n \sin{\left(x \right)} \cos{\left(n x \right)}}{n^{2} - 1} + \frac{3 \sin{\left(n x \right)} \cos{\left(x \right)}}{n^{2} - 1} & \text{otherwise} \end{cases}$$
The answer
[src]
/ 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.