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- Integral of d{x}:
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Integral of sin²xcos²x
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Integral of tg2x
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Integral of 1/(x^2-9)
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Identical expressions
- (four - two *x)*sin(kx)
- (4 minus 2 multiply by x) multiply by sinus of (kx)
- (four minus two multiply by x) multiply by sinus of (kx)
- (4-2x)sin(kx)
- 4-2xsinkx
- (4-2*x)*sin(kx)dx
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Similar expressions
- (4+2*x)*sin(kx)
Integral of (4-2*x)*sin(kx) dx
The solution
You have entered
[src]
pi / | | (4 - 2*x)*sin(k*x) dx | / 0
$$\int\limits_{0}^{\pi} \left(4 - 2 x\right) \sin{\left(k x \right)}\, dx$$
Integral((4 - 2*x)*sin(k*x), (x, 0, pi))
The answer (Indefinite)
[src]
// 0 for k = 0\
|| |
/ || //sin(k*x) \ | // 0 for k = 0\ // 0 for k = 0\
| || ||-------- for k != 0| | || | || |
| (4 - 2*x)*sin(k*x) dx = C + 2*|<-|< k | | + 4*|<-cos(k*x) | - 2*x*|<-cos(k*x) |
| || || | | ||---------- otherwise| ||---------- otherwise|
/ || \\ x otherwise / | \\ k / \\ k /
||------------------------- otherwise|
\\ k /
$$\int \left(4 - 2 x\right) \sin{\left(k x \right)}\, dx = C - 2 x \left(\begin{cases} 0 & \text{for}\: k = 0 \\- \frac{\cos{\left(k x \right)}}{k} & \text{otherwise} \end{cases}\right) + 2 \left(\begin{cases} 0 & \text{for}\: k = 0 \\- \frac{\begin{cases} \frac{\sin{\left(k x \right)}}{k} & \text{for}\: k \neq 0 \\x & \text{otherwise} \end{cases}}{k} & \text{otherwise} \end{cases}\right) + 4 \left(\begin{cases} 0 & \text{for}\: k = 0 \\- \frac{\cos{\left(k x \right)}}{k} & \text{otherwise} \end{cases}\right)$$
The answer
[src]
/4 4*cos(pi*k) 2*sin(pi*k) 2*pi*cos(pi*k) |- - ----------- - ----------- + -------------- for And(k > -oo, k < oo, k != 0) |k k 2 k < k | | 0 otherwise \
$$\begin{cases} - \frac{4 \cos{\left(\pi k \right)}}{k} + \frac{2 \pi \cos{\left(\pi k \right)}}{k} + \frac{4}{k} - \frac{2 \sin{\left(\pi k \right)}}{k^{2}} & \text{for}\: k > -\infty \wedge k < \infty \wedge k \neq 0 \\0 & \text{otherwise} \end{cases}$$
=
=
/4 4*cos(pi*k) 2*sin(pi*k) 2*pi*cos(pi*k) |- - ----------- - ----------- + -------------- for And(k > -oo, k < oo, k != 0) |k k 2 k < k | | 0 otherwise \
$$\begin{cases} - \frac{4 \cos{\left(\pi k \right)}}{k} + \frac{2 \pi \cos{\left(\pi k \right)}}{k} + \frac{4}{k} - \frac{2 \sin{\left(\pi k \right)}}{k^{2}} & \text{for}\: k > -\infty \wedge k < \infty \wedge k \neq 0 \\0 & \text{otherwise} \end{cases}$$
Piecewise((4/k - 4*cos(pi*k)/k - 2*sin(pi*k)/k^2 + 2*pi*cos(pi*k)/k, (k > -oo)∧(k < oo)∧(Ne(k, 0))), (0, True))
Use the examples entering the upper and lower limits of integration.