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- Improper integral
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How to use it?
- Integral of d{x}:
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Integral of x^e
- Integral of 1/(x^2+2x)
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Integral of dx/(4x-1)
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Integral of (3-x^2)^3
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Identical expressions
- (pi-x/ two)*cos(n*x)
- ( Pi minus x divide by 2) multiply by co sinus of e of (n multiply by x)
- ( Pi minus x divide by two) multiply by co sinus of e of (n multiply by x)
- (pi-x/2)cos(nx)
- pi-x/2cosnx
- (pi-x divide by 2)*cos(n*x)
- (pi-x/2)*cos(n*x)dx
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Similar expressions
- (pi+x/2)*cos(n*x)
Integral of (pi-x/2)*cos(n*x) dx
The solution
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pi -- 3 / | | / x\ | |pi - -|*cos(n*x) dx | \ 2/ | / pi
$$\int\limits_{\pi}^{\frac{\pi}{3}} \left(- \frac{x}{2} + \pi\right) \cos{\left(n x \right)}\, dx$$
Integral((pi - x/2)*cos(n*x), (x, pi, pi/3))
The answer (Indefinite)
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/ 2
| x
| -- for n = 0
| 2
|
|/-cos(n*x)
<|---------- for n != 0
|< n
|| // x for n = 0\
|\ 0 otherwise || |
/ |----------------------- otherwise x*|
$$\int \left(- \frac{x}{2} + \pi\right) \cos{\left(n x \right)}\, dx = C - \frac{x \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{\sin{\left(n x \right)}}{n} & \text{otherwise} \end{cases}\right)}{2} + \pi \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{\sin{\left(n x \right)}}{n} & \text{otherwise} \end{cases}\right) + \frac{\begin{cases} \frac{x^{2}}{2} & \text{for}\: n = 0 \\\frac{\begin{cases} - \frac{\cos{\left(n x \right)}}{n} & \text{for}\: n \neq 0 \\0 & \text{otherwise} \end{cases}}{n} & \text{otherwise} \end{cases}}{2}$$
The answer
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/ /pi*n\ /pi*n\ | cos|----| 5*pi*sin|----| |cos(pi*n) \ 3 / pi*sin(pi*n) \ 3 / |--------- - --------- - ------------ + -------------- for And(n > -oo, n < oo, n != 0) | 2 2 2*n 6*n < 2*n 2*n | | 2 | -4*pi | ------ otherwise \ 9
$$\begin{cases} \frac{5 \pi \sin{\left(\frac{\pi n}{3} \right)}}{6 n} - \frac{\pi \sin{\left(\pi n \right)}}{2 n} - \frac{\cos{\left(\frac{\pi n}{3} \right)}}{2 n^{2}} + \frac{\cos{\left(\pi n \right)}}{2 n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\- \frac{4 \pi^{2}}{9} & \text{otherwise} \end{cases}$$
=
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/ /pi*n\ /pi*n\ | cos|----| 5*pi*sin|----| |cos(pi*n) \ 3 / pi*sin(pi*n) \ 3 / |--------- - --------- - ------------ + -------------- for And(n > -oo, n < oo, n != 0) | 2 2 2*n 6*n < 2*n 2*n | | 2 | -4*pi | ------ otherwise \ 9
$$\begin{cases} \frac{5 \pi \sin{\left(\frac{\pi n}{3} \right)}}{6 n} - \frac{\pi \sin{\left(\pi n \right)}}{2 n} - \frac{\cos{\left(\frac{\pi n}{3} \right)}}{2 n^{2}} + \frac{\cos{\left(\pi n \right)}}{2 n^{2}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\- \frac{4 \pi^{2}}{9} & \text{otherwise} \end{cases}$$
Piecewise((cos(pi*n)/(2*n^2) - cos(pi*n/3)/(2*n^2) - pi*sin(pi*n)/(2*n) + 5*pi*sin(pi*n/3)/(6*n), (n > -oo)∧(n < oo)∧(Ne(n, 0))), (-4*pi^2/9, True))
Use the examples entering the upper and lower limits of integration.