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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^x*cos(x)
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Integral of cosx/sqrt(sinx)
- Integral of (dx)/((2x^2)+(8x)+(20))
- Integral of sin(ax)*cos(bx)
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Identical expressions
- sin(ax)*cos(bx)
- sinus of (ax) multiply by co sinus of e of (bx)
- sin(ax)cos(bx)
- sinaxcosbx
- sin(ax)*cos(bx)dx
Integral of sin(ax)*cos(bx) dx
The solution
You have entered
[src]
1 / | | sin(a*x)*cos(b*x) dx | / 0
$$\int\limits_{0}^{1} \sin{\left(a x \right)} \cos{\left(b x \right)}\, dx$$
The answer (Indefinite)
[src]
// 0 for And(a = 0, b = 0)\
|| |
|| 2 |
|| cos (b*x) |
|| --------- for a = -b |
|| 2*b |
/ || |
| || 2 |
| sin(a*x)*cos(b*x) dx = C + |< -cos (b*x) |
| || ----------- for a = b |
/ || 2*b |
|| |
|| a*cos(a*x)*cos(b*x) b*sin(a*x)*sin(b*x) |
||- ------------------- - ------------------- otherwise |
|| 2 2 2 2 |
|| a - b a - b |
\\ /
$$-{{\cos \left(\left(b+a\right)\,x\right)}\over{2\,\left(b+a\right)
}}-{{\cos \left(\left(a-b\right)\,x\right)}\over{2\,\left(a-b\right)
}}$$
The answer
[src]
/ 0 for Or(And(a = 0, b = 0), And(a = 0, a = b, b = 0), And(a = 0, a = -b, b = 0), And(a = 0, a = -b, a = b, b = 0)) | | 2 | 1 cos (b) | - --- + ------- for Or(And(a = 0, a = -b), And(a = -b, a = b), And(a = -b, b = 0), And(a = 0, a = -b, a = b), And(a = -b, a = b, b = 0), a = -b) | 2*b 2*b | | 2 < 1 cos (b) | --- - ------- for Or(And(a = 0, a = b), And(a = b, b = 0), a = b) | 2*b 2*b | | a a*cos(a)*cos(b) b*sin(a)*sin(b) |------- - --------------- - --------------- otherwise | 2 2 2 2 2 2 |a - b a - b a - b \
$$-{{\left(b-a\right)\,\cos \left(b+a\right)+\left(-b-a\right)\,\cos
\left(b-a\right)}\over{2\,b^2-2\,a^2}}-{{a}\over{b^2-a^2}}$$
=
=
/ 0 for Or(And(a = 0, b = 0), And(a = 0, a = b, b = 0), And(a = 0, a = -b, b = 0), And(a = 0, a = -b, a = b, b = 0)) | | 2 | 1 cos (b) | - --- + ------- for Or(And(a = 0, a = -b), And(a = -b, a = b), And(a = -b, b = 0), And(a = 0, a = -b, a = b), And(a = -b, a = b, b = 0), a = -b) | 2*b 2*b | | 2 < 1 cos (b) | --- - ------- for Or(And(a = 0, a = b), And(a = b, b = 0), a = b) | 2*b 2*b | | a a*cos(a)*cos(b) b*sin(a)*sin(b) |------- - --------------- - --------------- otherwise | 2 2 2 2 2 2 |a - b a - b a - b \
$$\begin{cases} 0 & \text{for}\: \left(a = 0 \wedge b = 0\right) \vee \left(a = 0 \wedge a = b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge a = b \wedge b = 0\right) \\\frac{\cos^{2}{\left(b \right)}}{2 b} - \frac{1}{2 b} & \text{for}\: \left(a = 0 \wedge a = - b\right) \vee \left(a = - b \wedge a = b\right) \vee \left(a = - b \wedge b = 0\right) \vee \left(a = 0 \wedge a = - b \wedge a = b\right) \vee \left(a = - b \wedge a = b \wedge b = 0\right) \vee a = - b \\- \frac{\cos^{2}{\left(b \right)}}{2 b} + \frac{1}{2 b} & \text{for}\: \left(a = 0 \wedge a = b\right) \vee \left(a = b \wedge b = 0\right) \vee a = b \\- \frac{a \cos{\left(a \right)} \cos{\left(b \right)}}{a^{2} - b^{2}} - \frac{b \sin{\left(a \right)} \sin{\left(b \right)}}{a^{2} - b^{2}} + \frac{a}{a^{2} - b^{2}} & \text{otherwise} \end{cases}$$
Use the examples entering the upper and lower limits of integration.