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- Integral of d{x}:
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Integral of x/(sinx^2)^2
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Integral of e^xsin(x/2)
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Integral of e^(-x)*cos(3*x)
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Identical expressions
- one /(two +sin(x)*sin(x))
- 1 divide by (2 plus sinus of (x) multiply by sinus of (x))
- one divide by (two plus sinus of (x) multiply by sinus of (x))
- 1/(2+sin(x)sin(x))
- 1/2+sinxsinx
- 1 divide by (2+sin(x)*sin(x))
- 1/(2+sin(x)*sin(x))dx
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Similar expressions
- 1/(2-sin(x)*sin(x))
- 1/(2+sinx*sinx)
Integral of 1/(2+sin(x)*sin(x)) dx
The solution
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pi -- 4 / | | 1 | ----------------- dx | 2 + sin(x)*sin(x) | / 0
$$\int\limits_{0}^{\frac{\pi}{4}} \frac{1}{\sin{\left(x \right)} \sin{\left(x \right)} + 2}\, dx$$
Integral(1/(2 + sin(x)*sin(x)), (x, 0, pi/4))
The answer (Indefinite)
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/ /x pi\ / /x\ \\ / /x pi\ / /x\ \\ / /x pi\ / /x\ \\
___________ | |- - --| | tan|-| || ___________ | |- - --| | tan|-| || ___________ | |- - --| | tan|-| ||
/ ___ | |2 2 | | \2/ || / ___ | |2 2 | | \2/ || ___ / ___ | |2 2 | | \2/ ||
\/ 2 - \/ 3 *|pi*floor|------| + atan|--------------|| 2*\/ 2 + \/ 3 *|pi*floor|------| + atan|--------------|| \/ 3 *\/ 2 + \/ 3 *|pi*floor|------| + atan|--------------||
/ | \ pi / | ___________|| | \ pi / | ___________|| | \ pi / | ___________||
| | | / ___ || | | / ___ || | | / ___ ||
| 1 \ \\/ 2 + \/ 3 // \ \\/ 2 - \/ 3 // \ \\/ 2 - \/ 3 //
| ----------------- dx = C - ----------------------------------------------------------------------- - ----------------------------------------------------------------------- + -----------------------------------------------------------------------
| 2 + sin(x)*sin(x) ___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________
| / ___ / ___ ___ / ___ / ___ / ___ / ___ ___ / ___ / ___ / ___ / ___ ___ / ___ / ___
/ - 3*\/ 2 + \/ 3 *\/ 2 - \/ 3 + \/ 3 *\/ 2 + \/ 3 *\/ 2 - \/ 3 - 3*\/ 2 + \/ 3 *\/ 2 - \/ 3 + \/ 3 *\/ 2 + \/ 3 *\/ 2 - \/ 3 - 3*\/ 2 + \/ 3 *\/ 2 - \/ 3 + \/ 3 *\/ 2 + \/ 3 *\/ 2 - \/ 3
$$\int \frac{1}{\sin{\left(x \right)} \sin{\left(x \right)} + 2}\, dx = C + \frac{\sqrt{3} \sqrt{\sqrt{3} + 2} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{2 - \sqrt{3}}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{- 3 \sqrt{2 - \sqrt{3}} \sqrt{\sqrt{3} + 2} + \sqrt{3} \sqrt{2 - \sqrt{3}} \sqrt{\sqrt{3} + 2}} - \frac{2 \sqrt{\sqrt{3} + 2} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{2 - \sqrt{3}}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{- 3 \sqrt{2 - \sqrt{3}} \sqrt{\sqrt{3} + 2} + \sqrt{3} \sqrt{2 - \sqrt{3}} \sqrt{\sqrt{3} + 2}} - \frac{\sqrt{2 - \sqrt{3}} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{\sqrt{3} + 2}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{- 3 \sqrt{2 - \sqrt{3}} \sqrt{\sqrt{3} + 2} + \sqrt{3} \sqrt{2 - \sqrt{3}} \sqrt{\sqrt{3} + 2}}$$
The graph
The answer
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/ / ___ \\ / / ___ \\ ___________ ___________ / / ___ \\ ___________ ___________ / / ___ \\
| | -1 + \/ 2 || ___ | | -1 + \/ 2 || / ___ / ___ | | -1 + \/ 2 || ___ / ___ / ___ | | -1 + \/ 2 ||
2*|-pi + atan|--------------|| \/ 3 *|-pi + atan|--------------|| 2*\/ 2 + \/ 3 *\/ 2 - \/ 3 *|-pi + atan|--------------|| \/ 3 *\/ 2 + \/ 3 *\/ 2 - \/ 3 *|-pi + atan|--------------||
| | ___________|| | | ___________|| ___________ ___________ | | ___________|| ___________ ___________ | | ___________||
| | / ___ || ___ | | / ___ || / ___ / ___ | | / ___ || ___ / ___ / ___ | | / ___ ||
2*pi \ \\/ 2 + \/ 3 // pi*\/ 3 \ \\/ 2 + \/ 3 // 2*pi*\/ 2 + \/ 3 *\/ 2 - \/ 3 \ \\/ 2 - \/ 3 // pi*\/ 3 *\/ 2 + \/ 3 *\/ 2 - \/ 3 \ \\/ 2 - \/ 3 //
- ------------------------------------------- - ------------------------------------------- + ------------------------------------------- + ------------------------------------------- - ------------------------------------------- - ------------------------------------------------------------ + ------------------------------------------- + ----------------------------------------------------------------
___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________
/ ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___
- 9*\/ 2 + \/ 3 + 5*\/ 3 *\/ 2 + \/ 3 - 9*\/ 2 + \/ 3 + 5*\/ 3 *\/ 2 + \/ 3 - 9*\/ 2 + \/ 3 + 5*\/ 3 *\/ 2 + \/ 3 - 9*\/ 2 + \/ 3 + 5*\/ 3 *\/ 2 + \/ 3 - 9*\/ 2 + \/ 3 + 5*\/ 3 *\/ 2 + \/ 3 - 9*\/ 2 + \/ 3 + 5*\/ 3 *\/ 2 + \/ 3 - 9*\/ 2 + \/ 3 + 5*\/ 3 *\/ 2 + \/ 3 - 9*\/ 2 + \/ 3 + 5*\/ 3 *\/ 2 + \/ 3
$$- \frac{2 \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{\sqrt{3} + 2}} \right)}\right)}{- 9 \sqrt{\sqrt{3} + 2} + 5 \sqrt{3} \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \pi}{- 9 \sqrt{\sqrt{3} + 2} + 5 \sqrt{3} \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \pi \sqrt{2 - \sqrt{3}} \sqrt{\sqrt{3} + 2}}{- 9 \sqrt{\sqrt{3} + 2} + 5 \sqrt{3} \sqrt{\sqrt{3} + 2}} - \frac{2 \sqrt{2 - \sqrt{3}} \sqrt{\sqrt{3} + 2} \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{2 - \sqrt{3}}} \right)}\right)}{- 9 \sqrt{\sqrt{3} + 2} + 5 \sqrt{3} \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \sqrt{2 - \sqrt{3}} \sqrt{\sqrt{3} + 2} \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{2 - \sqrt{3}}} \right)}\right)}{- 9 \sqrt{\sqrt{3} + 2} + 5 \sqrt{3} \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{\sqrt{3} + 2}} \right)}\right)}{- 9 \sqrt{\sqrt{3} + 2} + 5 \sqrt{3} \sqrt{\sqrt{3} + 2}} - \frac{2 \pi}{- 9 \sqrt{\sqrt{3} + 2} + 5 \sqrt{3} \sqrt{\sqrt{3} + 2}} - \frac{2 \pi \sqrt{2 - \sqrt{3}} \sqrt{\sqrt{3} + 2}}{- 9 \sqrt{\sqrt{3} + 2} + 5 \sqrt{3} \sqrt{\sqrt{3} + 2}}$$
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/ / ___ \\ / / ___ \\ ___________ ___________ / / ___ \\ ___________ ___________ / / ___ \\
| | -1 + \/ 2 || ___ | | -1 + \/ 2 || / ___ / ___ | | -1 + \/ 2 || ___ / ___ / ___ | | -1 + \/ 2 ||
2*|-pi + atan|--------------|| \/ 3 *|-pi + atan|--------------|| 2*\/ 2 + \/ 3 *\/ 2 - \/ 3 *|-pi + atan|--------------|| \/ 3 *\/ 2 + \/ 3 *\/ 2 - \/ 3 *|-pi + atan|--------------||
| | ___________|| | | ___________|| ___________ ___________ | | ___________|| ___________ ___________ | | ___________||
| | / ___ || ___ | | / ___ || / ___ / ___ | | / ___ || ___ / ___ / ___ | | / ___ ||
2*pi \ \\/ 2 + \/ 3 // pi*\/ 3 \ \\/ 2 + \/ 3 // 2*pi*\/ 2 + \/ 3 *\/ 2 - \/ 3 \ \\/ 2 - \/ 3 // pi*\/ 3 *\/ 2 + \/ 3 *\/ 2 - \/ 3 \ \\/ 2 - \/ 3 //
- ------------------------------------------- - ------------------------------------------- + ------------------------------------------- + ------------------------------------------- - ------------------------------------------- - ------------------------------------------------------------ + ------------------------------------------- + ----------------------------------------------------------------
___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________ ___________
/ ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___
- 9*\/ 2 + \/ 3 + 5*\/ 3 *\/ 2 + \/ 3 - 9*\/ 2 + \/ 3 + 5*\/ 3 *\/ 2 + \/ 3 - 9*\/ 2 + \/ 3 + 5*\/ 3 *\/ 2 + \/ 3 - 9*\/ 2 + \/ 3 + 5*\/ 3 *\/ 2 + \/ 3 - 9*\/ 2 + \/ 3 + 5*\/ 3 *\/ 2 + \/ 3 - 9*\/ 2 + \/ 3 + 5*\/ 3 *\/ 2 + \/ 3 - 9*\/ 2 + \/ 3 + 5*\/ 3 *\/ 2 + \/ 3 - 9*\/ 2 + \/ 3 + 5*\/ 3 *\/ 2 + \/ 3
$$- \frac{2 \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{\sqrt{3} + 2}} \right)}\right)}{- 9 \sqrt{\sqrt{3} + 2} + 5 \sqrt{3} \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \pi}{- 9 \sqrt{\sqrt{3} + 2} + 5 \sqrt{3} \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \pi \sqrt{2 - \sqrt{3}} \sqrt{\sqrt{3} + 2}}{- 9 \sqrt{\sqrt{3} + 2} + 5 \sqrt{3} \sqrt{\sqrt{3} + 2}} - \frac{2 \sqrt{2 - \sqrt{3}} \sqrt{\sqrt{3} + 2} \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{2 - \sqrt{3}}} \right)}\right)}{- 9 \sqrt{\sqrt{3} + 2} + 5 \sqrt{3} \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \sqrt{2 - \sqrt{3}} \sqrt{\sqrt{3} + 2} \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{2 - \sqrt{3}}} \right)}\right)}{- 9 \sqrt{\sqrt{3} + 2} + 5 \sqrt{3} \sqrt{\sqrt{3} + 2}} + \frac{\sqrt{3} \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{\sqrt{3} + 2}} \right)}\right)}{- 9 \sqrt{\sqrt{3} + 2} + 5 \sqrt{3} \sqrt{\sqrt{3} + 2}} - \frac{2 \pi}{- 9 \sqrt{\sqrt{3} + 2} + 5 \sqrt{3} \sqrt{\sqrt{3} + 2}} - \frac{2 \pi \sqrt{2 - \sqrt{3}} \sqrt{\sqrt{3} + 2}}{- 9 \sqrt{\sqrt{3} + 2} + 5 \sqrt{3} \sqrt{\sqrt{3} + 2}}$$
-2*pi/(-9*sqrt(2 + sqrt(3)) + 5*sqrt(3)*sqrt(2 + sqrt(3))) - 2*(-pi + atan((-1 + sqrt(2))/sqrt(2 + sqrt(3))))/(-9*sqrt(2 + sqrt(3)) + 5*sqrt(3)*sqrt(2 + sqrt(3))) + pi*sqrt(3)/(-9*sqrt(2 + sqrt(3)) + 5*sqrt(3)*sqrt(2 + sqrt(3))) + sqrt(3)*(-pi + atan((-1 + sqrt(2))/sqrt(2 + sqrt(3))))/(-9*sqrt(2 + sqrt(3)) + 5*sqrt(3)*sqrt(2 + sqrt(3))) - 2*pi*sqrt(2 + sqrt(3))*sqrt(2 - sqrt(3))/(-9*sqrt(2 + sqrt(3)) + 5*sqrt(3)*sqrt(2 + sqrt(3))) - 2*sqrt(2 + sqrt(3))*sqrt(2 - sqrt(3))*(-pi + atan((-1 + sqrt(2))/sqrt(2 - sqrt(3))))/(-9*sqrt(2 + sqrt(3)) + 5*sqrt(3)*sqrt(2 + sqrt(3))) + pi*sqrt(3)*sqrt(2 + sqrt(3))*sqrt(2 - sqrt(3))/(-9*sqrt(2 + sqrt(3)) + 5*sqrt(3)*sqrt(2 + sqrt(3))) + sqrt(3)*sqrt(2 + sqrt(3))*sqrt(2 - sqrt(3))*(-pi + atan((-1 + sqrt(2))/sqrt(2 - sqrt(3))))/(-9*sqrt(2 + sqrt(3)) + 5*sqrt(3)*sqrt(2 + sqrt(3)))
Use the examples entering the upper and lower limits of integration.