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Integral of x/(3x²-1)^4
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Integral of √(9-x^2)
- 12*y*sin(2*x*y)
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Identical expressions
- twelve *y*sin(two *x*y)
- 12 multiply by y multiply by sinus of (2 multiply by x multiply by y)
- twelve multiply by y multiply by sinus of (two multiply by x multiply by y)
- 12ysin(2xy)
- 12ysin2xy
- 12*y*sin(2*x*y)dx
Integral of 12*y*sin(2*x*y) dy
The solution
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pi -- 2 / | | 12*y*sin(2*x*y) dy | / pi -- 4
$$\int\limits_{\frac{\pi}{4}}^{\frac{\pi}{2}} 12 y \sin{\left(2 x y \right)}\, dy$$
Integral((12*y)*sin((2*x)*y), (y, pi/4, pi/2))
The answer (Indefinite)
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// 0 for x = 0\
|| |
/ || //sin(2*x*y) \ | // 0 for x = 0\
| || ||---------- for 2*x != 0| | || |
| 12*y*sin(2*x*y) dy = C - 12*|<-|< 2*x | | + 12*y*|<-cos(2*x*y) |
| || || | | ||------------ otherwise|
/ || \\ y otherwise / | \\ 2*x /
||----------------------------- otherwise|
\\ 2*x /
$$\int 12 y \sin{\left(2 x y \right)}\, dy = C + 12 y \left(\begin{cases} 0 & \text{for}\: x = 0 \\- \frac{\cos{\left(2 x y \right)}}{2 x} & \text{otherwise} \end{cases}\right) - 12 \left(\begin{cases} 0 & \text{for}\: x = 0 \\- \frac{\begin{cases} \frac{\sin{\left(2 x y \right)}}{2 x} & \text{for}\: 2 x \neq 0 \\y & \text{otherwise} \end{cases}}{2 x} & \text{otherwise} \end{cases}\right)$$
The answer
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/ /pi*x\ /pi*x\ | 3*sin|----| 3*pi*cos|----| | \ 2 / 3*sin(pi*x) 3*pi*cos(pi*x) \ 2 / |- ----------- + ----------- - -------------- + -------------- for And(x > -oo, x < oo, x != 0) < 2 2 x 2*x | x x | | 0 otherwise \
$$\begin{cases} \frac{3 \pi \cos{\left(\frac{\pi x}{2} \right)}}{2 x} - \frac{3 \pi \cos{\left(\pi x \right)}}{x} - \frac{3 \sin{\left(\frac{\pi x}{2} \right)}}{x^{2}} + \frac{3 \sin{\left(\pi x \right)}}{x^{2}} & \text{for}\: x > -\infty \wedge x < \infty \wedge x \neq 0 \\0 & \text{otherwise} \end{cases}$$
=
=
/ /pi*x\ /pi*x\ | 3*sin|----| 3*pi*cos|----| | \ 2 / 3*sin(pi*x) 3*pi*cos(pi*x) \ 2 / |- ----------- + ----------- - -------------- + -------------- for And(x > -oo, x < oo, x != 0) < 2 2 x 2*x | x x | | 0 otherwise \
$$\begin{cases} \frac{3 \pi \cos{\left(\frac{\pi x}{2} \right)}}{2 x} - \frac{3 \pi \cos{\left(\pi x \right)}}{x} - \frac{3 \sin{\left(\frac{\pi x}{2} \right)}}{x^{2}} + \frac{3 \sin{\left(\pi x \right)}}{x^{2}} & \text{for}\: x > -\infty \wedge x < \infty \wedge x \neq 0 \\0 & \text{otherwise} \end{cases}$$
Piecewise((-3*sin(pi*x/2)/x^2 + 3*sin(pi*x)/x^2 - 3*pi*cos(pi*x)/x + 3*pi*cos(pi*x/2)/(2*x), (x > -oo)∧(x < oo)∧(Ne(x, 0))), (0, True))
Use the examples entering the upper and lower limits of integration.