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- Improper integral
- Double Integral
- Triple Integral
- Curvilinear integral
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- Differential equations Step by Step
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How to use it?
- Integral of d{x}:
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Integral of √(1+x²)
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Integral of ln(x^2)
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Integral of x^2dx
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Integral of e^(5*x)
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Identical expressions
- dy/(y^ two - two)
- dy divide by (y squared minus 2)
- dy divide by (y to the power of two minus two)
- dy/(y2-2)
- dy/y2-2
- dy/(y²-2)
- dy/(y to the power of 2-2)
- dy/y^2-2
- dy divide by (y^2-2)
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Similar expressions
- dy/(y^2+2)
Integral of dy/(y^2-2) dx
The solution
You have entered
[src]
1 / | | 1 | ------ dy | 2 | y - 2 | / 0
$$\int\limits_{0}^{1} \frac{1}{y^{2} - 2}\, dy$$
Integral(1/(y^2 - 2), (y, 0, 1))
Detail solution
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Add the constant of integration:
PiecewiseRule(subfunctions=[(ArctanRule(a=1, b=1, c=-2, context=1/(y**2 - 2), symbol=y), False), (ArccothRule(a=1, b=1, c=-2, context=1/(y**2 - 2), symbol=y), y**2 > 2), (ArctanhRule(a=1, b=1, c=-2, context=1/(y**2 - 2), symbol=y), y**2 < 2)], context=1/(y**2 - 2), symbol=y)
The answer is:
The answer (Indefinite)
[src]
// / ___\ \
|| ___ |y*\/ 2 | |
||-\/ 2 *acoth|-------| |
/ || \ 2 / 2 |
| ||---------------------- for y > 2|
| 1 || 2 |
| ------ dy = C + |< |
| 2 || / ___\ |
| y - 2 || ___ |y*\/ 2 | |
| ||-\/ 2 *atanh|-------| |
/ || \ 2 / 2 |
||---------------------- for y < 2|
\\ 2 /
$$\int \frac{1}{y^{2} - 2}\, dy = C + \begin{cases} - \frac{\sqrt{2} \operatorname{acoth}{\left(\frac{\sqrt{2} y}{2} \right)}}{2} & \text{for}\: y^{2} > 2 \\- \frac{\sqrt{2} \operatorname{atanh}{\left(\frac{\sqrt{2} y}{2} \right)}}{2} & \text{for}\: y^{2} < 2 \end{cases}$$
The graph
The answer
[src]
___ / / ___\\ ___ / ___\ ___ / / ___\\ ___ / ___\
\/ 2 *\pi*I + log\\/ 2 // \/ 2 *log\1 + \/ 2 / \/ 2 *\pi*I + log\-1 + \/ 2 // \/ 2 *log\\/ 2 /
- ------------------------- - -------------------- + ------------------------------ + ----------------
4 4 4 4
$$- \frac{\sqrt{2} \log{\left(1 + \sqrt{2} \right)}}{4} + \frac{\sqrt{2} \log{\left(\sqrt{2} \right)}}{4} - \frac{\sqrt{2} \left(\log{\left(\sqrt{2} \right)} + i \pi\right)}{4} + \frac{\sqrt{2} \left(\log{\left(-1 + \sqrt{2} \right)} + i \pi\right)}{4}$$
=
=
___ / / ___\\ ___ / ___\ ___ / / ___\\ ___ / ___\
\/ 2 *\pi*I + log\\/ 2 // \/ 2 *log\1 + \/ 2 / \/ 2 *\pi*I + log\-1 + \/ 2 // \/ 2 *log\\/ 2 /
- ------------------------- - -------------------- + ------------------------------ + ----------------
4 4 4 4
$$- \frac{\sqrt{2} \log{\left(1 + \sqrt{2} \right)}}{4} + \frac{\sqrt{2} \log{\left(\sqrt{2} \right)}}{4} - \frac{\sqrt{2} \left(\log{\left(\sqrt{2} \right)} + i \pi\right)}{4} + \frac{\sqrt{2} \left(\log{\left(-1 + \sqrt{2} \right)} + i \pi\right)}{4}$$
-sqrt(2)*(pi*i + log(sqrt(2)))/4 - sqrt(2)*log(1 + sqrt(2))/4 + sqrt(2)*(pi*i + log(-1 + sqrt(2)))/4 + sqrt(2)*log(sqrt(2))/4
Use the examples entering the upper and lower limits of integration.