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- Integral of d{x}:
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Integral of 2x(x^2+1)
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Integral of (1+2x^2)/(x^2(1+x^2))
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Integral of 1/(2+cosx)^2
- Integral of x/((x-1)(x-2))
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Identical expressions
- one /((chx)^ two +(shx)^ two)
- 1 divide by ((chx) squared plus (shx) squared )
- one divide by ((chx) to the power of two plus (shx) to the power of two)
- 1/((chx)2+(shx)2)
- 1/chx2+shx2
- 1/((chx)²+(shx)²)
- 1/((chx) to the power of 2+(shx) to the power of 2)
- 1/chx^2+shx^2
- 1 divide by ((chx)^2+(shx)^2)
- 1/((chx)^2+(shx)^2)dx
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Similar expressions
- 1/((chx)^2-(shx)^2)
Integral of 1/((chx)^2+(shx)^2) dx
The solution
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[src]
1 / | | 1 | ------------------- dx | 2 2 | cosh (x) + sinh (x) | / 0
$$\int\limits_{0}^{1} \frac{1}{\sinh^{2}{\left(x \right)} + \cosh^{2}{\left(x \right)}}\, dx$$
Integral(1/(cosh(x)^2 + sinh(x)^2), (x, 0, 1))
The answer (Indefinite)
[src]
/ /x\ \ / /x\ \ / /x\ \ / /x\ \
| tanh|-| | | tanh|-| | _____________ _____________ | tanh|-| | _____________ _____________ | tanh|-| |
| \2/ | ___ | \2/ | / ___ / ___ | \2/ | ___ / ___ / ___ | \2/ |
3*atan|----------------| 2*\/ 2 *atan|----------------| 3*\/ 3 - 2*\/ 2 *\/ 3 + 2*\/ 2 *atan|----------------| 2*\/ 2 *\/ 3 - 2*\/ 2 *\/ 3 + 2*\/ 2 *atan|----------------|
/ | _____________| | _____________| | _____________| | _____________|
| | / ___ | | / ___ | | / ___ | | / ___ |
| 1 \\/ 3 - 2*\/ 2 / \\/ 3 - 2*\/ 2 / \\/ 3 + 2*\/ 2 / \\/ 3 + 2*\/ 2 /
| ------------------- dx = C + --------------------------------------------- + --------------------------------------------- - ---------------------------------------------------------- - ----------------------------------------------------------------
| 2 2 _____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________
| cosh (x) + sinh (x) / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___
| 7*\/ 3 - 2*\/ 2 + 5*\/ 2 *\/ 3 - 2*\/ 2 7*\/ 3 - 2*\/ 2 + 5*\/ 2 *\/ 3 - 2*\/ 2 7*\/ 3 - 2*\/ 2 + 5*\/ 2 *\/ 3 - 2*\/ 2 7*\/ 3 - 2*\/ 2 + 5*\/ 2 *\/ 3 - 2*\/ 2
/
$$\int \frac{1}{\sinh^{2}{\left(x \right)} + \cosh^{2}{\left(x \right)}}\, dx = C + \frac{2 \sqrt{2} \operatorname{atan}{\left(\frac{\tanh{\left(\frac{x}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)}}{7 \sqrt{3 - 2 \sqrt{2}} + 5 \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} + \frac{3 \operatorname{atan}{\left(\frac{\tanh{\left(\frac{x}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)}}{7 \sqrt{3 - 2 \sqrt{2}} + 5 \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} - \frac{3 \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \operatorname{atan}{\left(\frac{\tanh{\left(\frac{x}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)}}{7 \sqrt{3 - 2 \sqrt{2}} + 5 \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} - \frac{2 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \operatorname{atan}{\left(\frac{\tanh{\left(\frac{x}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)}}{7 \sqrt{3 - 2 \sqrt{2}} + 5 \sqrt{2} \sqrt{3 - 2 \sqrt{2}}}$$
The graph
The answer
[src]
_____________ _____________ _____________ _____________
/ tanh(1/2) \ ___ / tanh(1/2) \ / ___ / ___ / tanh(1/2) \ ___ / ___ / ___ / tanh(1/2) \
3*atan|----------------| 2*\/ 2 *atan|----------------| 3*\/ 3 - 2*\/ 2 *\/ 3 + 2*\/ 2 *atan|----------------| 2*\/ 2 *\/ 3 - 2*\/ 2 *\/ 3 + 2*\/ 2 *atan|----------------|
| _____________| | _____________| | _____________| | _____________|
| / ___ | | / ___ | | / ___ | | / ___ |
\\/ 3 - 2*\/ 2 / \\/ 3 - 2*\/ 2 / \\/ 3 + 2*\/ 2 / \\/ 3 + 2*\/ 2 /
--------------------------------------------- + --------------------------------------------- - ---------------------------------------------------------- - ----------------------------------------------------------------
_____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________
/ ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___
7*\/ 3 - 2*\/ 2 + 5*\/ 2 *\/ 3 - 2*\/ 2 7*\/ 3 - 2*\/ 2 + 5*\/ 2 *\/ 3 - 2*\/ 2 7*\/ 3 - 2*\/ 2 + 5*\/ 2 *\/ 3 - 2*\/ 2 7*\/ 3 - 2*\/ 2 + 5*\/ 2 *\/ 3 - 2*\/ 2
$$- \frac{3 \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \operatorname{atan}{\left(\frac{\tanh{\left(\frac{1}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)}}{7 \sqrt{3 - 2 \sqrt{2}} + 5 \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} - \frac{2 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \operatorname{atan}{\left(\frac{\tanh{\left(\frac{1}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)}}{7 \sqrt{3 - 2 \sqrt{2}} + 5 \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} + \frac{2 \sqrt{2} \operatorname{atan}{\left(\frac{\tanh{\left(\frac{1}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)}}{7 \sqrt{3 - 2 \sqrt{2}} + 5 \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} + \frac{3 \operatorname{atan}{\left(\frac{\tanh{\left(\frac{1}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)}}{7 \sqrt{3 - 2 \sqrt{2}} + 5 \sqrt{2} \sqrt{3 - 2 \sqrt{2}}}$$
=
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_____________ _____________ _____________ _____________
/ tanh(1/2) \ ___ / tanh(1/2) \ / ___ / ___ / tanh(1/2) \ ___ / ___ / ___ / tanh(1/2) \
3*atan|----------------| 2*\/ 2 *atan|----------------| 3*\/ 3 - 2*\/ 2 *\/ 3 + 2*\/ 2 *atan|----------------| 2*\/ 2 *\/ 3 - 2*\/ 2 *\/ 3 + 2*\/ 2 *atan|----------------|
| _____________| | _____________| | _____________| | _____________|
| / ___ | | / ___ | | / ___ | | / ___ |
\\/ 3 - 2*\/ 2 / \\/ 3 - 2*\/ 2 / \\/ 3 + 2*\/ 2 / \\/ 3 + 2*\/ 2 /
--------------------------------------------- + --------------------------------------------- - ---------------------------------------------------------- - ----------------------------------------------------------------
_____________ _____________ _____________ _____________ _____________ _____________ _____________ _____________
/ ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___ / ___ ___ / ___
7*\/ 3 - 2*\/ 2 + 5*\/ 2 *\/ 3 - 2*\/ 2 7*\/ 3 - 2*\/ 2 + 5*\/ 2 *\/ 3 - 2*\/ 2 7*\/ 3 - 2*\/ 2 + 5*\/ 2 *\/ 3 - 2*\/ 2 7*\/ 3 - 2*\/ 2 + 5*\/ 2 *\/ 3 - 2*\/ 2
$$- \frac{3 \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \operatorname{atan}{\left(\frac{\tanh{\left(\frac{1}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)}}{7 \sqrt{3 - 2 \sqrt{2}} + 5 \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} - \frac{2 \sqrt{2} \sqrt{3 - 2 \sqrt{2}} \sqrt{2 \sqrt{2} + 3} \operatorname{atan}{\left(\frac{\tanh{\left(\frac{1}{2} \right)}}{\sqrt{2 \sqrt{2} + 3}} \right)}}{7 \sqrt{3 - 2 \sqrt{2}} + 5 \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} + \frac{2 \sqrt{2} \operatorname{atan}{\left(\frac{\tanh{\left(\frac{1}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)}}{7 \sqrt{3 - 2 \sqrt{2}} + 5 \sqrt{2} \sqrt{3 - 2 \sqrt{2}}} + \frac{3 \operatorname{atan}{\left(\frac{\tanh{\left(\frac{1}{2} \right)}}{\sqrt{3 - 2 \sqrt{2}}} \right)}}{7 \sqrt{3 - 2 \sqrt{2}} + 5 \sqrt{2} \sqrt{3 - 2 \sqrt{2}}}$$
3*atan(tanh(1/2)/sqrt(3 - 2*sqrt(2)))/(7*sqrt(3 - 2*sqrt(2)) + 5*sqrt(2)*sqrt(3 - 2*sqrt(2))) + 2*sqrt(2)*atan(tanh(1/2)/sqrt(3 - 2*sqrt(2)))/(7*sqrt(3 - 2*sqrt(2)) + 5*sqrt(2)*sqrt(3 - 2*sqrt(2))) - 3*sqrt(3 - 2*sqrt(2))*sqrt(3 + 2*sqrt(2))*atan(tanh(1/2)/sqrt(3 + 2*sqrt(2)))/(7*sqrt(3 - 2*sqrt(2)) + 5*sqrt(2)*sqrt(3 - 2*sqrt(2))) - 2*sqrt(2)*sqrt(3 - 2*sqrt(2))*sqrt(3 + 2*sqrt(2))*atan(tanh(1/2)/sqrt(3 + 2*sqrt(2)))/(7*sqrt(3 - 2*sqrt(2)) + 5*sqrt(2)*sqrt(3 - 2*sqrt(2)))
Use the examples entering the upper and lower limits of integration.