Mister Exam
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- Improper integral
- Double Integral
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How to use it?
- Integral of d{x}:
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Integral of e^(2*x)*dx
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Integral of (x+2)e^(-x)
- Integral of (tanx)^1/2
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Integral of sin(x)*cos(2x)
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Identical expressions
- sin(x)*sin(n*x)
- sinus of (x) multiply by sinus of (n multiply by x)
- sin(x)sin(nx)
- sinxsinnx
- sin(x)*sin(n*x)dx
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Similar expressions
- sin(x)sin(nx)
- sinx*sin(n*x)
Integral of sin(x)*sin(n*x) dx
The solution
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[src]
1 / | | sin(x)*sin(n*x) dx | / 0
$$\int\limits_{0}^{1} \sin{\left(x \right)} \sin{\left(n x \right)}\, dx$$
Integral(sin(x)*sin(n*x), (x, 0, 1))
The answer (Indefinite)
[src]
// 2 2 \
||cos(x)*sin(x) x*cos (x) x*sin (x) |
||------------- - --------- - --------- for n = -1|
|| 2 2 2 |
|| |
/ || 2 2 |
| ||x*cos (x) x*sin (x) cos(x)*sin(x) |
| sin(x)*sin(n*x) dx = C + |<--------- + --------- - ------------- for n = 1 |
| || 2 2 2 |
/ || |
|| cos(x)*sin(n*x) n*cos(n*x)*sin(x) |
|| --------------- - ----------------- otherwise |
|| 2 2 |
|| -1 + n -1 + n |
\\ /
$$\int \sin{\left(x \right)} \sin{\left(n x \right)}\, dx = C + \begin{cases} - \frac{x \sin^{2}{\left(x \right)}}{2} - \frac{x \cos^{2}{\left(x \right)}}{2} + \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{2} & \text{for}\: n = -1 \\\frac{x \sin^{2}{\left(x \right)}}{2} + \frac{x \cos^{2}{\left(x \right)}}{2} - \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{2} & \text{for}\: n = 1 \\- \frac{n \sin{\left(x \right)} \cos{\left(n x \right)}}{n^{2} - 1} + \frac{\sin{\left(n x \right)} \cos{\left(x \right)}}{n^{2} - 1} & \text{otherwise} \end{cases}$$
The answer
[src]
/ 2 2 | cos (1) sin (1) cos(1)*sin(1) |- ------- - ------- + ------------- for n = -1 | 2 2 2 | | 2 2 | cos (1) sin (1) cos(1)*sin(1) < ------- + ------- - ------------- for n = 1 | 2 2 2 | | cos(1)*sin(n) n*cos(n)*sin(1) | ------------- - --------------- otherwise | 2 2 | -1 + n -1 + n \
$$\begin{cases} - \frac{\sin^{2}{\left(1 \right)}}{2} - \frac{\cos^{2}{\left(1 \right)}}{2} + \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} & \text{for}\: n = -1 \\- \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{\cos^{2}{\left(1 \right)}}{2} + \frac{\sin^{2}{\left(1 \right)}}{2} & \text{for}\: n = 1 \\- \frac{n \sin{\left(1 \right)} \cos{\left(n \right)}}{n^{2} - 1} + \frac{\sin{\left(n \right)} \cos{\left(1 \right)}}{n^{2} - 1} & \text{otherwise} \end{cases}$$
=
=
/ 2 2 | cos (1) sin (1) cos(1)*sin(1) |- ------- - ------- + ------------- for n = -1 | 2 2 2 | | 2 2 | cos (1) sin (1) cos(1)*sin(1) < ------- + ------- - ------------- for n = 1 | 2 2 2 | | cos(1)*sin(n) n*cos(n)*sin(1) | ------------- - --------------- otherwise | 2 2 | -1 + n -1 + n \
$$\begin{cases} - \frac{\sin^{2}{\left(1 \right)}}{2} - \frac{\cos^{2}{\left(1 \right)}}{2} + \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} & \text{for}\: n = -1 \\- \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{\cos^{2}{\left(1 \right)}}{2} + \frac{\sin^{2}{\left(1 \right)}}{2} & \text{for}\: n = 1 \\- \frac{n \sin{\left(1 \right)} \cos{\left(n \right)}}{n^{2} - 1} + \frac{\sin{\left(n \right)} \cos{\left(1 \right)}}{n^{2} - 1} & \text{otherwise} \end{cases}$$
Piecewise((-cos(1)^2/2 - sin(1)^2/2 + cos(1)*sin(1)/2, n = -1), (cos(1)^2/2 + sin(1)^2/2 - cos(1)*sin(1)/2, n = 1), (cos(1)*sin(n)/(-1 + n^2) - n*cos(n)*sin(1)/(-1 + n^2), True))
Use the examples entering the upper and lower limits of integration.