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- Integral of d{x}:
- Integral of x+y
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Integral of t
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Integral of atan(x)
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Identical expressions
- one /(one +sin^ two (x))
- 1 divide by (1 plus sinus of squared (x))
- one divide by (one plus sinus of to the power of two (x))
- 1/(1+sin2(x))
- 1/1+sin2x
- 1/(1+sin²(x))
- 1/(1+sin to the power of 2(x))
- 1/1+sin^2x
- 1 divide by (1+sin^2(x))
- 1/(1+sin^2(x))dx
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Similar expressions
- 1/(1-sin^2(x))
Integral of 1/(1+sin^2(x)) dx
The solution
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[src]
pi -- 4 / | | 1 | ----------- dx | 2 | 1 + sin (x) | / 0
$$\int\limits_{0}^{\frac{\pi}{4}} \frac{1}{\sin^{2}{\left(x \right)} + 1}\, dx$$
Integral(1/(1 + sin(x)^2), (x, 0, pi/4))
The answer (Indefinite)
[src]
/x pi\ / /x\ \ / /x pi\ / /x\ \\
|- - --| | tan|-| | _____________ _____________ | |- - --| | tan|-| ||
|2 2 | | \2/ | / ___ / ___ | |2 2 | | \2/ ||
pi*floor|------| + atan|----------------| \/ 3 - 2*\/ 2 *\/ 3 + 2*\/ 2 *|pi*floor|------| + atan|----------------||
/ \ pi / | _____________| | \ pi / | _____________||
| | / ___ | | | / ___ ||
| 1 \\/ 3 - 2*\/ 2 / \ \\/ 3 + 2*\/ 2 //
| ----------- dx = C + ------------------------------------------- + -----------------------------------------------------------------------------
| 2 _____________ _____________ _____________ _____________
| 1 + sin (x) / ___ ___ / ___ / ___ ___ / ___
| 2*\/ 3 - 2*\/ 2 + \/ 2 *\/ 3 - 2*\/ 2 2*\/ 3 - 2*\/ 2 + \/ 2 *\/ 3 - 2*\/ 2
/
$$\int \frac{1}{\sin^{2}{\left(x \right)} + 1}\, dx = C + \frac{\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor}{\sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 2 \sqrt{3 - 2 \sqrt{2}}} + \frac{\sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \left(\operatorname{atan}{\left(\frac{\tan{\left(\frac{x}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)} + \pi \left\lfloor{\frac{\frac{x}{2} - \frac{\pi}{2}}{\pi}}\right\rfloor\right)}{\sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 2 \sqrt{3 - 2 \sqrt{2}}}$$
The graph
The answer
[src]
/ / ___ \\ / / ___ \\ _____________ _____________ / / ___ \\ _____________ _____________ / / ___ \\
| | -1 + \/ 2 || ___ | | -1 + \/ 2 || / ___ / ___ | | -1 + \/ 2 || ___ / ___ / ___ | | -1 + \/ 2 ||
3*|-pi + atan|----------------|| 2*\/ 2 *|-pi + atan|----------------|| 3*\/ 3 - 2*\/ 2 *\/ 3 + 2*\/ 2 *|-pi + atan|----------------|| 2*\/ 2 *\/ 3 - 2*\/ 2 *\/ 3 + 2*\/ 2 *|-pi + atan|----------------||
| | _____________|| | | _____________|| _____________ _____________ | | _____________|| _____________ _____________ | | _____________||
| | / ___ || ___ | | / ___ || / ___ / ___ | | / ___ || ___ / ___ / ___ | | / ___ ||
3*pi \ \\/ 3 - 2*\/ 2 // 2*pi*\/ 2 \ \\/ 3 - 2*\/ 2 // 3*pi*\/ 3 - 2*\/ 2 *\/ 3 + 2*\/ 2 \ \\/ 3 + 2*\/ 2 // 2*pi*\/ 2 *\/ 3 - 2*\/ 2 *\/ 3 + 2*\/ 2 \ \\/ 3 + 2*\/ 2 //
---------------------------------------------- + ---------------------------------------------- + ---------------------------------------------- + ---------------------------------------------- + ---------------------------------------------- + ------------------------------------------------------------------ + ---------------------------------------------- + ------------------------------------------------------------------------
_____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________
/ ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___
10*\/ 3 - 2*\/ 2 + 7*\/ 2 *\/ 3 - 2*\/ 2 10*\/ 3 - 2*\/ 2 + 7*\/ 2 *\/ 3 - 2*\/ 2 10*\/ 3 - 2*\/ 2 + 7*\/ 2 *\/ 3 - 2*\/ 2 10*\/ 3 - 2*\/ 2 + 7*\/ 2 *\/ 3 - 2*\/ 2 10*\/ 3 - 2*\/ 2 + 7*\/ 2 *\/ 3 - 2*\/ 2 10*\/ 3 - 2*\/ 2 + 7*\/ 2 *\/ 3 - 2*\/ 2 10*\/ 3 - 2*\/ 2 + 7*\/ 2 *\/ 3 - 2*\/ 2 10*\/ 3 - 2*\/ 2 + 7*\/ 2 *\/ 3 - 2*\/ 2
$$\frac{3 \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{2 \sqrt{2} + 3}} \right)}\right)}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 10 \sqrt{3 - 2 \sqrt{2}}} + \frac{2 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{2 \sqrt{2} + 3}} \right)}\right)}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 10 \sqrt{3 - 2 \sqrt{2}}} + \frac{3 \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{3 - 2 \sqrt{2}}} \right)}\right)}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 10 \sqrt{3 - 2 \sqrt{2}}} + \frac{2 \sqrt{2} \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{3 - 2 \sqrt{2}}} \right)}\right)}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 10 \sqrt{3 - 2 \sqrt{2}}} + \frac{2 \sqrt{2} \pi}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 10 \sqrt{3 - 2 \sqrt{2}}} + \frac{2 \sqrt{2} \pi \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3}}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 10 \sqrt{3 - 2 \sqrt{2}}} + \frac{3 \pi}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 10 \sqrt{3 - 2 \sqrt{2}}} + \frac{3 \pi \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3}}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 10 \sqrt{3 - 2 \sqrt{2}}}$$
=
=
/ / ___ \\ / / ___ \\ _____________ _____________ / / ___ \\ _____________ _____________ / / ___ \\
| | -1 + \/ 2 || ___ | | -1 + \/ 2 || / ___ / ___ | | -1 + \/ 2 || ___ / ___ / ___ | | -1 + \/ 2 ||
3*|-pi + atan|----------------|| 2*\/ 2 *|-pi + atan|----------------|| 3*\/ 3 - 2*\/ 2 *\/ 3 + 2*\/ 2 *|-pi + atan|----------------|| 2*\/ 2 *\/ 3 - 2*\/ 2 *\/ 3 + 2*\/ 2 *|-pi + atan|----------------||
| | _____________|| | | _____________|| _____________ _____________ | | _____________|| _____________ _____________ | | _____________||
| | / ___ || ___ | | / ___ || / ___ / ___ | | / ___ || ___ / ___ / ___ | | / ___ ||
3*pi \ \\/ 3 - 2*\/ 2 // 2*pi*\/ 2 \ \\/ 3 - 2*\/ 2 // 3*pi*\/ 3 - 2*\/ 2 *\/ 3 + 2*\/ 2 \ \\/ 3 + 2*\/ 2 // 2*pi*\/ 2 *\/ 3 - 2*\/ 2 *\/ 3 + 2*\/ 2 \ \\/ 3 + 2*\/ 2 //
---------------------------------------------- + ---------------------------------------------- + ---------------------------------------------- + ---------------------------------------------- + ---------------------------------------------- + ------------------------------------------------------------------ + ---------------------------------------------- + ------------------------------------------------------------------------
_____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________
/ ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___
10*\/ 3 - 2*\/ 2 + 7*\/ 2 *\/ 3 - 2*\/ 2 10*\/ 3 - 2*\/ 2 + 7*\/ 2 *\/ 3 - 2*\/ 2 10*\/ 3 - 2*\/ 2 + 7*\/ 2 *\/ 3 - 2*\/ 2 10*\/ 3 - 2*\/ 2 + 7*\/ 2 *\/ 3 - 2*\/ 2 10*\/ 3 - 2*\/ 2 + 7*\/ 2 *\/ 3 - 2*\/ 2 10*\/ 3 - 2*\/ 2 + 7*\/ 2 *\/ 3 - 2*\/ 2 10*\/ 3 - 2*\/ 2 + 7*\/ 2 *\/ 3 - 2*\/ 2 10*\/ 3 - 2*\/ 2 + 7*\/ 2 *\/ 3 - 2*\/ 2
$$\frac{3 \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{2 \sqrt{2} + 3}} \right)}\right)}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 10 \sqrt{3 - 2 \sqrt{2}}} + \frac{2 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{2 \sqrt{2} + 3}} \right)}\right)}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 10 \sqrt{3 - 2 \sqrt{2}}} + \frac{3 \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{3 - 2 \sqrt{2}}} \right)}\right)}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 10 \sqrt{3 - 2 \sqrt{2}}} + \frac{2 \sqrt{2} \left(- \pi + \operatorname{atan}{\left(\frac{-1 + \sqrt{2}}{\sqrt{3 - 2 \sqrt{2}}} \right)}\right)}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 10 \sqrt{3 - 2 \sqrt{2}}} + \frac{2 \sqrt{2} \pi}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 10 \sqrt{3 - 2 \sqrt{2}}} + \frac{2 \sqrt{2} \pi \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3}}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 10 \sqrt{3 - 2 \sqrt{2}}} + \frac{3 \pi}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 10 \sqrt{3 - 2 \sqrt{2}}} + \frac{3 \pi \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3}}{7 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} + 10 \sqrt{3 - 2 \sqrt{2}}}$$
3*pi/(10*sqrt(3 - 2*sqrt(2)) + 7*sqrt(2)*sqrt(3 - 2*sqrt(2))) + 3*(-pi + atan((-1 + sqrt(2))/sqrt(3 - 2*sqrt(2))))/(10*sqrt(3 - 2*sqrt(2)) + 7*sqrt(2)*sqrt(3 - 2*sqrt(2))) + 2*pi*sqrt(2)/(10*sqrt(3 - 2*sqrt(2)) + 7*sqrt(2)*sqrt(3 - 2*sqrt(2))) + 2*sqrt(2)*(-pi + atan((-1 + sqrt(2))/sqrt(3 - 2*sqrt(2))))/(10*sqrt(3 - 2*sqrt(2)) + 7*sqrt(2)*sqrt(3 - 2*sqrt(2))) + 3*pi*sqrt(3 - 2*sqrt(2))*sqrt(3 + 2*sqrt(2))/(10*sqrt(3 - 2*sqrt(2)) + 7*sqrt(2)*sqrt(3 - 2*sqrt(2))) + 3*sqrt(3 - 2*sqrt(2))*sqrt(3 + 2*sqrt(2))*(-pi + atan((-1 + sqrt(2))/sqrt(3 + 2*sqrt(2))))/(10*sqrt(3 - 2*sqrt(2)) + 7*sqrt(2)*sqrt(3 - 2*sqrt(2))) + 2*pi*sqrt(2)*sqrt(3 - 2*sqrt(2))*sqrt(3 + 2*sqrt(2))/(10*sqrt(3 - 2*sqrt(2)) + 7*sqrt(2)*sqrt(3 - 2*sqrt(2))) + 2*sqrt(2)*sqrt(3 - 2*sqrt(2))*sqrt(3 + 2*sqrt(2))*(-pi + atan((-1 + sqrt(2))/sqrt(3 + 2*sqrt(2))))/(10*sqrt(3 - 2*sqrt(2)) + 7*sqrt(2)*sqrt(3 - 2*sqrt(2)))
Use the examples entering the upper and lower limits of integration.