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How to use it?
- Integral of d{x}:
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Integral of xe^(1-x)
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Integral of (x^3+2)/(x^3-4x)
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Integral of tang(x)
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Integral of x/(1-x^4)^(1/2)
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Identical expressions
- sin(w*t+f)^ two
- sinus of (w multiply by t plus f) squared
- sinus of (w multiply by t plus f) to the power of two
- sin(w*t+f)2
- sinw*t+f2
- sin(w*t+f)²
- sin(w*t+f) to the power of 2
- sin(wt+f)^2
- sin(wt+f)2
- sinwt+f2
- sinwt+f^2
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Similar expressions
- sin(w*t-f)^2
Integral of sin(w*t+f)^2 dx
The solution
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[src]
1 / | | 2 | sin (w*t + f) dw | / 0
$$\int\limits_{0}^{1} \sin^{2}{\left(f + t w \right)}\, dw$$
Integral(sin(w*t + f)^2, (w, 0, 1))
The answer (Indefinite)
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// /f t*w\ 3/f t*w\ 4/f t*w\ 2/f t*w\ \
|| 2*tan|- + ---| 2*tan |- + ---| t*w*tan |- + ---| 2*t*w*tan |- + ---| |
/ || \2 2 / \2 2 / t*w \2 2 / \2 2 / |
| ||- ------------------------------------------- + ------------------------------------------- + ------------------------------------------- + ------------------------------------------- + ------------------------------------------- for t != 0|
| 2 || 4/f t*w\ 2/f t*w\ 4/f t*w\ 2/f t*w\ 4/f t*w\ 2/f t*w\ 4/f t*w\ 2/f t*w\ 4/f t*w\ 2/f t*w\ |
| sin (w*t + f) dw = C + |< 2*t + 2*t*tan |- + ---| + 4*t*tan |- + ---| 2*t + 2*t*tan |- + ---| + 4*t*tan |- + ---| 2*t + 2*t*tan |- + ---| + 4*t*tan |- + ---| 2*t + 2*t*tan |- + ---| + 4*t*tan |- + ---| 2*t + 2*t*tan |- + ---| + 4*t*tan |- + ---| |
| || \2 2 / \2 2 / \2 2 / \2 2 / \2 2 / \2 2 / \2 2 / \2 2 / \2 2 / \2 2 / |
/ || |
|| 2 |
|| w*sin (f) otherwise |
\\ /
$$\int \sin^{2}{\left(f + t w \right)}\, dw = C + \begin{cases} \frac{t w \tan^{4}{\left(\frac{f}{2} + \frac{t w}{2} \right)}}{2 t \tan^{4}{\left(\frac{f}{2} + \frac{t w}{2} \right)} + 4 t \tan^{2}{\left(\frac{f}{2} + \frac{t w}{2} \right)} + 2 t} + \frac{2 t w \tan^{2}{\left(\frac{f}{2} + \frac{t w}{2} \right)}}{2 t \tan^{4}{\left(\frac{f}{2} + \frac{t w}{2} \right)} + 4 t \tan^{2}{\left(\frac{f}{2} + \frac{t w}{2} \right)} + 2 t} + \frac{t w}{2 t \tan^{4}{\left(\frac{f}{2} + \frac{t w}{2} \right)} + 4 t \tan^{2}{\left(\frac{f}{2} + \frac{t w}{2} \right)} + 2 t} + \frac{2 \tan^{3}{\left(\frac{f}{2} + \frac{t w}{2} \right)}}{2 t \tan^{4}{\left(\frac{f}{2} + \frac{t w}{2} \right)} + 4 t \tan^{2}{\left(\frac{f}{2} + \frac{t w}{2} \right)} + 2 t} - \frac{2 \tan{\left(\frac{f}{2} + \frac{t w}{2} \right)}}{2 t \tan^{4}{\left(\frac{f}{2} + \frac{t w}{2} \right)} + 4 t \tan^{2}{\left(\frac{f}{2} + \frac{t w}{2} \right)} + 2 t} & \text{for}\: t \neq 0 \\w \sin^{2}{\left(f \right)} & \text{otherwise} \end{cases}$$
The answer
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/ 2 2 |cos (f + t) sin (f + t) cos(f)*sin(f) cos(f + t)*sin(f + t) |----------- + ----------- + ------------- - --------------------- for And(t > -oo, t < oo, t != 0) < 2 2 2*t 2*t | | 2 \ sin (f) otherwise
$$\begin{cases} \frac{\sin^{2}{\left(f + t \right)}}{2} + \frac{\cos^{2}{\left(f + t \right)}}{2} + \frac{\sin{\left(f \right)} \cos{\left(f \right)}}{2 t} - \frac{\sin{\left(f + t \right)} \cos{\left(f + t \right)}}{2 t} & \text{for}\: t > -\infty \wedge t < \infty \wedge t \neq 0 \\\sin^{2}{\left(f \right)} & \text{otherwise} \end{cases}$$
=
=
/ 2 2 |cos (f + t) sin (f + t) cos(f)*sin(f) cos(f + t)*sin(f + t) |----------- + ----------- + ------------- - --------------------- for And(t > -oo, t < oo, t != 0) < 2 2 2*t 2*t | | 2 \ sin (f) otherwise
$$\begin{cases} \frac{\sin^{2}{\left(f + t \right)}}{2} + \frac{\cos^{2}{\left(f + t \right)}}{2} + \frac{\sin{\left(f \right)} \cos{\left(f \right)}}{2 t} - \frac{\sin{\left(f + t \right)} \cos{\left(f + t \right)}}{2 t} & \text{for}\: t > -\infty \wedge t < \infty \wedge t \neq 0 \\\sin^{2}{\left(f \right)} & \text{otherwise} \end{cases}$$
Piecewise((cos(f + t)^2/2 + sin(f + t)^2/2 + cos(f)*sin(f)/(2*t) - cos(f + t)*sin(f + t)/(2*t), (t > -oo)∧(t < oo)∧(Ne(t, 0))), (sin(f)^2, True))
Use the examples entering the upper and lower limits of integration.