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- Integral of d{x}:
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Integral of 1/(2x-1)
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Integral of (4-x^2)^(1/2)
- Integral of x^2/(x-1)
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Integral of (x^2)/(x+1)
- Canonical form:
- sqrt(x^2-y^2)
- The double integral of:
- sqrt(x^2-y^2)
- sqrt(x^2-y^2)
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Identical expressions
- sqrt(x^ two -y^ two)
- square root of (x squared minus y squared )
- square root of (x to the power of two minus y to the power of two)
- √(x^2-y^2)
- sqrt(x2-y2)
- sqrtx2-y2
- sqrt(x²-y²)
- sqrt(x to the power of 2-y to the power of 2)
- sqrtx^2-y^2
- sqrt(x^2-y^2)dx
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Similar expressions
- sqrt(x^2+y^2)
Integral of sqrt(x^2-y^2) dx
The solution
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[src]
0 / | | _________ | / 2 2 | \/ x - y dx | / -r
$$\int\limits_{- r}^{0} \sqrt{x^{2} - y^{2}}\, dx$$
Integral(sqrt(x^2 - y^2), (x, -r, 0))
The answer (Indefinite)
[src]
// 2 /x\ \
|| y *acosh|-| 3 | 2| |
|| \y/ x x*y |x | |
||- ----------- + ------------------- - ----------------- for |--| > 1|
|| 2 _________ _________ | 2| |
/ || / 2 / 2 |y | |
| || / x / x |
| _________ || 2*y* / -1 + -- 2* / -1 + -- |
| / 2 2 || / 2 / 2 |
| \/ x - y dx = C + |< \/ y \/ y |
| || |
/ || ________ |
|| / 2 |
|| / x |
|| 2 /x\ I*x*y* / 1 - -- |
|| I*y *asin|-| / 2 |
|| \y/ \/ y |
|| ------------ + -------------------- otherwise |
\\ 2 2 /
$$\int \sqrt{x^{2} - y^{2}}\, dx = C + \begin{cases} \frac{x^{3}}{2 y \sqrt{\frac{x^{2}}{y^{2}} - 1}} - \frac{x y}{2 \sqrt{\frac{x^{2}}{y^{2}} - 1}} - \frac{y^{2} \operatorname{acosh}{\left(\frac{x}{y} \right)}}{2} & \text{for}\: \left|{\frac{x^{2}}{y^{2}}}\right| > 1 \\\frac{i x y \sqrt{- \frac{x^{2}}{y^{2}} + 1}}{2} + \frac{i y^{2} \operatorname{asin}{\left(\frac{x}{y} \right)}}{2} & \text{otherwise} \end{cases}$$
The answer
[src]
/ 0 | / | | | | / 2 4 2 | 2| | | | y x x 3*x |x | | | |- --------------- + ---------------- - ----------------- + ------------------- for |--| > 1 | | | _________ 3/2 3/2 _________ | 2| | | | / 2 / 2\ / 2\ / 2 |y | | | | / x | x | 3 | x | / x | | | / -1 + -- 2*y*|-1 + --| 2*y *|-1 + --| 2*y* / -1 + -- | | | / 2 | 2| | 2| / 2 | | | \/ y \ y / \ y / \/ y | | | | | | ________ | | < / 2 dx for r > 0 | | | / x | | | I*y* / 1 - -- | | | / 2 2 | | | \/ y I*y I*x | | | ------------------ + ---------------- - ------------------ otherwise | | | 2 ________ ________ | | | / 2 / 2 | | | / x / x | | | 2* / 1 - -- 2*y* / 1 - -- | | | / 2 / 2 | | \ \/ y \/ y | | |/ |-r < | -r | / | | | | / 2 4 2 | 2| | | | y x x 3*x |x | | | |- --------------- + ---------------- - ----------------- + ------------------- for |--| > 1 | | | _________ 3/2 3/2 _________ | 2| | | | / 2 / 2\ / 2\ / 2 |y | | | | / x | x | 3 | x | / x | | | / -1 + -- 2*y*|-1 + --| 2*y *|-1 + --| 2*y* / -1 + -- | | | / 2 | 2| | 2| / 2 | | | \/ y \ y / \ y / \/ y | | | | | | ________ |- | < / 2 dx otherwise | | | / x | | | I*y* / 1 - -- | | | / 2 2 | | | \/ y I*y I*x | | | ------------------ + ---------------- - ------------------ otherwise | | | 2 ________ ________ | | | / 2 / 2 | | | / x / x | | | 2* / 1 - -- 2*y* / 1 - -- | | | / 2 / 2 | | \ \/ y \/ y | | | / \ 0
$$\begin{cases} \int\limits_{- r}^{0} \begin{cases} - \frac{x^{4}}{2 y^{3} \left(\frac{x^{2}}{y^{2}} - 1\right)^{\frac{3}{2}}} + \frac{3 x^{2}}{2 y \sqrt{\frac{x^{2}}{y^{2}} - 1}} + \frac{x^{2}}{2 y \left(\frac{x^{2}}{y^{2}} - 1\right)^{\frac{3}{2}}} - \frac{y}{\sqrt{\frac{x^{2}}{y^{2}} - 1}} & \text{for}\: \left|{\frac{x^{2}}{y^{2}}}\right| > 1 \\- \frac{i x^{2}}{2 y \sqrt{- \frac{x^{2}}{y^{2}} + 1}} + \frac{i y \sqrt{- \frac{x^{2}}{y^{2}} + 1}}{2} + \frac{i y}{2 \sqrt{- \frac{x^{2}}{y^{2}} + 1}} & \text{otherwise} \end{cases}\, dx & \text{for}\: r > 0 \\- \int\limits_{0}^{- r} \begin{cases} - \frac{x^{4}}{2 y^{3} \left(\frac{x^{2}}{y^{2}} - 1\right)^{\frac{3}{2}}} + \frac{3 x^{2}}{2 y \sqrt{\frac{x^{2}}{y^{2}} - 1}} + \frac{x^{2}}{2 y \left(\frac{x^{2}}{y^{2}} - 1\right)^{\frac{3}{2}}} - \frac{y}{\sqrt{\frac{x^{2}}{y^{2}} - 1}} & \text{for}\: \left|{\frac{x^{2}}{y^{2}}}\right| > 1 \\- \frac{i x^{2}}{2 y \sqrt{- \frac{x^{2}}{y^{2}} + 1}} + \frac{i y \sqrt{- \frac{x^{2}}{y^{2}} + 1}}{2} + \frac{i y}{2 \sqrt{- \frac{x^{2}}{y^{2}} + 1}} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases}$$
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/ 0 | / | | | | / 2 4 2 | 2| | | | y x x 3*x |x | | | |- --------------- + ---------------- - ----------------- + ------------------- for |--| > 1 | | | _________ 3/2 3/2 _________ | 2| | | | / 2 / 2\ / 2\ / 2 |y | | | | / x | x | 3 | x | / x | | | / -1 + -- 2*y*|-1 + --| 2*y *|-1 + --| 2*y* / -1 + -- | | | / 2 | 2| | 2| / 2 | | | \/ y \ y / \ y / \/ y | | | | | | ________ | | < / 2 dx for r > 0 | | | / x | | | I*y* / 1 - -- | | | / 2 2 | | | \/ y I*y I*x | | | ------------------ + ---------------- - ------------------ otherwise | | | 2 ________ ________ | | | / 2 / 2 | | | / x / x | | | 2* / 1 - -- 2*y* / 1 - -- | | | / 2 / 2 | | \ \/ y \/ y | | |/ |-r < | -r | / | | | | / 2 4 2 | 2| | | | y x x 3*x |x | | | |- --------------- + ---------------- - ----------------- + ------------------- for |--| > 1 | | | _________ 3/2 3/2 _________ | 2| | | | / 2 / 2\ / 2\ / 2 |y | | | | / x | x | 3 | x | / x | | | / -1 + -- 2*y*|-1 + --| 2*y *|-1 + --| 2*y* / -1 + -- | | | / 2 | 2| | 2| / 2 | | | \/ y \ y / \ y / \/ y | | | | | | ________ |- | < / 2 dx otherwise | | | / x | | | I*y* / 1 - -- | | | / 2 2 | | | \/ y I*y I*x | | | ------------------ + ---------------- - ------------------ otherwise | | | 2 ________ ________ | | | / 2 / 2 | | | / x / x | | | 2* / 1 - -- 2*y* / 1 - -- | | | / 2 / 2 | | \ \/ y \/ y | | | / \ 0
$$\begin{cases} \int\limits_{- r}^{0} \begin{cases} - \frac{x^{4}}{2 y^{3} \left(\frac{x^{2}}{y^{2}} - 1\right)^{\frac{3}{2}}} + \frac{3 x^{2}}{2 y \sqrt{\frac{x^{2}}{y^{2}} - 1}} + \frac{x^{2}}{2 y \left(\frac{x^{2}}{y^{2}} - 1\right)^{\frac{3}{2}}} - \frac{y}{\sqrt{\frac{x^{2}}{y^{2}} - 1}} & \text{for}\: \left|{\frac{x^{2}}{y^{2}}}\right| > 1 \\- \frac{i x^{2}}{2 y \sqrt{- \frac{x^{2}}{y^{2}} + 1}} + \frac{i y \sqrt{- \frac{x^{2}}{y^{2}} + 1}}{2} + \frac{i y}{2 \sqrt{- \frac{x^{2}}{y^{2}} + 1}} & \text{otherwise} \end{cases}\, dx & \text{for}\: r > 0 \\- \int\limits_{0}^{- r} \begin{cases} - \frac{x^{4}}{2 y^{3} \left(\frac{x^{2}}{y^{2}} - 1\right)^{\frac{3}{2}}} + \frac{3 x^{2}}{2 y \sqrt{\frac{x^{2}}{y^{2}} - 1}} + \frac{x^{2}}{2 y \left(\frac{x^{2}}{y^{2}} - 1\right)^{\frac{3}{2}}} - \frac{y}{\sqrt{\frac{x^{2}}{y^{2}} - 1}} & \text{for}\: \left|{\frac{x^{2}}{y^{2}}}\right| > 1 \\- \frac{i x^{2}}{2 y \sqrt{- \frac{x^{2}}{y^{2}} + 1}} + \frac{i y \sqrt{- \frac{x^{2}}{y^{2}} + 1}}{2} + \frac{i y}{2 \sqrt{- \frac{x^{2}}{y^{2}} + 1}} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases}$$
Piecewise((Integral(Piecewise((-y/sqrt(-1 + x^2/y^2) + x^2/(2*y*(-1 + x^2/y^2)^(3/2)) - x^4/(2*y^3*(-1 + x^2/y^2)^(3/2)) + 3*x^2/(2*y*sqrt(-1 + x^2/y^2)), |x^2/y^2| > 1), (i*y*sqrt(1 - x^2/y^2)/2 + i*y/(2*sqrt(1 - x^2/y^2)) - i*x^2/(2*y*sqrt(1 - x^2/y^2)), True)), (x, -r, 0)), r > 0), (-Integral(Piecewise((-y/sqrt(-1 + x^2/y^2) + x^2/(2*y*(-1 + x^2/y^2)^(3/2)) - x^4/(2*y^3*(-1 + x^2/y^2)^(3/2)) + 3*x^2/(2*y*sqrt(-1 + x^2/y^2)), |x^2/y^2| > 1), (i*y*sqrt(1 - x^2/y^2)/2 + i*y/(2*sqrt(1 - x^2/y^2)) - i*x^2/(2*y*sqrt(1 - x^2/y^2)), True)), (x, 0, -r)), True))
Use the examples entering the upper and lower limits of integration.