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- Integral of d{x}:
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Integral of x^2*e^3x
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Integral of x/(2x+3)
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Integral of x^2/sqrt(x^2-25)
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Integral of x^2sqrt(x^2-9)
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Identical expressions
- (one /pi)2x*cos(ax)
- (1 divide by Pi )2x multiply by co sinus of e of (ax)
- (one divide by Pi )2x multiply by co sinus of e of (ax)
- (1/pi)2xcos(ax)
- 1/pi2xcosax
- (1 divide by pi)2x*cos(ax)
- (1/pi)2x*cos(ax)dx
Integral of (1/pi)2x*cos(ax) dx
The solution
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0 / | | 2 | --*x*cos(a*x) dx | pi | / -pi
$$\int\limits_{- \pi}^{0} x \frac{2}{\pi} \cos{\left(a x \right)}\, dx$$
Integral(((2/pi)*x)*cos(a*x), (x, -pi, 0))
The answer (Indefinite)
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// 2 \
|| x |
|| -- for a = 0|
|| 2 |
|| |
||/-cos(a*x) |
2*|<|---------- for a != 0 |
||< a |
||| | // x for a = 0\
||\ 0 otherwise | || |
/ ||----------------------- otherwise| 2*x*|
$$\int x \frac{2}{\pi} \cos{\left(a x \right)}\, dx = C + \frac{2 x \left(\begin{cases} x & \text{for}\: a = 0 \\\frac{\sin{\left(a x \right)}}{a} & \text{otherwise} \end{cases}\right)}{\pi} - \frac{2 \left(\begin{cases} \frac{x^{2}}{2} & \text{for}\: a = 0 \\\frac{\begin{cases} - \frac{\cos{\left(a x \right)}}{a} & \text{for}\: a \neq 0 \\0 & \text{otherwise} \end{cases}}{a} & \text{otherwise} \end{cases}\right)}{\pi}$$
The answer
[src]
/ /cos(pi*a) pi*sin(pi*a)\ | 2*|--------- + ------------| | | 2 a | | \ a / 2 <- ---------------------------- + ----- for And(a > -oo, a < oo, a != 0) | pi 2 | pi*a | \ -pi otherwise
$$\begin{cases} - \frac{2 \left(\frac{\pi \sin{\left(\pi a \right)}}{a} + \frac{\cos{\left(\pi a \right)}}{a^{2}}\right)}{\pi} + \frac{2}{\pi a^{2}} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\- \pi & \text{otherwise} \end{cases}$$
=
=
/ /cos(pi*a) pi*sin(pi*a)\ | 2*|--------- + ------------| | | 2 a | | \ a / 2 <- ---------------------------- + ----- for And(a > -oo, a < oo, a != 0) | pi 2 | pi*a | \ -pi otherwise
$$\begin{cases} - \frac{2 \left(\frac{\pi \sin{\left(\pi a \right)}}{a} + \frac{\cos{\left(\pi a \right)}}{a^{2}}\right)}{\pi} + \frac{2}{\pi a^{2}} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\- \pi & \text{otherwise} \end{cases}$$
Piecewise((-2*(cos(pi*a)/a^2 + pi*sin(pi*a)/a)/pi + 2/(pi*a^2), (a > -oo)∧(a < oo)∧(Ne(a, 0))), (-pi, True))
Use the examples entering the upper and lower limits of integration.