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Identical expressions
- sqrt(y^ two -x^ two)
- square root of (y squared minus x squared )
- square root of (y to the power of two minus x to the power of two)
- √(y^2-x^2)
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- sqrty2-x2
- sqrt(y²-x²)
- sqrt(y to the power of 2-x to the power of 2)
- sqrty^2-x^2
- sqrt(y^2-x^2)dx
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Similar expressions
- sqrt(y^2+x^2)
Integral of sqrt(y^2-x^2) dx
The solution
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[src]
sin(x)
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0
$$\int\limits_{0}^{\sin{\left(x \right)}} \sqrt{- x^{2} + y^{2}}\, dx$$
Integral(sqrt(y^2 - x^2), (x, 0, sin(x)))
The answer (Indefinite)
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// 2 /x\ \
|| I*y *acosh|-| 3 | 2| |
|| \y/ I*x I*x*y |x | |
||- ------------- + ------------------- - ----------------- for |--| > 1|
|| 2 _________ _________ | 2| |
/ || / 2 / 2 |y | |
| || / x / x |
| _________ || 2*y* / -1 + -- 2* / -1 + -- |
| / 2 2 || / 2 / 2 |
| \/ y - x dx = C + |< \/ y \/ y |
| || |
/ || ________ |
|| / 2 |
|| / x |
|| 2 /x\ x*y* / 1 - -- |
|| y *asin|-| / 2 |
|| \y/ \/ y |
|| ---------- + ------------------ otherwise |
\\ 2 2 /
$$\int \sqrt{- x^{2} + y^{2}}\, dx = C + \begin{cases} \frac{i x^{3}}{2 y \sqrt{\frac{x^{2}}{y^{2}} - 1}} - \frac{i x y}{2 \sqrt{\frac{x^{2}}{y^{2}} - 1}} - \frac{i y^{2} \operatorname{acosh}{\left(\frac{x}{y} \right)}}{2} & \text{for}\: \left|{\frac{x^{2}}{y^{2}}}\right| > 1 \\\frac{x y \sqrt{- \frac{x^{2}}{y^{2}} + 1}}{2} + \frac{y^{2} \operatorname{asin}{\left(\frac{x}{y} \right)}}{2} & \text{otherwise} \end{cases}$$
The answer
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/ 0 | / | | | | / 2 4 2 | 2| | | | I*y I*x I*x 3*I*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 sin(x) < 0 | | | / x | | | y* / 1 - -- | | | / 2 2 | | | \/ y y x | | | ---------------- + ---------------- - ------------------ otherwise | | | 2 ________ ________ | | | / 2 / 2 | | | / x / x | | | 2* / 1 - -- 2*y* / 1 - -- | | | / 2 / 2 | | \ \/ y \/ y | | | / | sin(x) < | sin(x) | / | | | | / 2 4 2 | 2| | | | I*y I*x I*x 3*I*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 | | | y* / 1 - -- | | | / 2 2 | | | \/ y y x | | | ---------------- + ---------------- - ------------------ otherwise | | | 2 ________ ________ | | | / 2 / 2 | | | / x / x | | | 2* / 1 - -- 2*y* / 1 - -- | | | / 2 / 2 | | \ \/ y \/ y | | | / \ 0
$$\begin{cases} - \int\limits_{\sin{\left(x \right)}}^{0} \begin{cases} - \frac{i x^{4}}{2 y^{3} \left(\frac{x^{2}}{y^{2}} - 1\right)^{\frac{3}{2}}} + \frac{3 i x^{2}}{2 y \sqrt{\frac{x^{2}}{y^{2}} - 1}} + \frac{i x^{2}}{2 y \left(\frac{x^{2}}{y^{2}} - 1\right)^{\frac{3}{2}}} - \frac{i y}{\sqrt{\frac{x^{2}}{y^{2}} - 1}} & \text{for}\: \left|{\frac{x^{2}}{y^{2}}}\right| > 1 \\- \frac{x^{2}}{2 y \sqrt{- \frac{x^{2}}{y^{2}} + 1}} + \frac{y \sqrt{- \frac{x^{2}}{y^{2}} + 1}}{2} + \frac{y}{2 \sqrt{- \frac{x^{2}}{y^{2}} + 1}} & \text{otherwise} \end{cases}\, dx & \text{for}\: \sin{\left(x \right)} < 0 \\\int\limits_{0}^{\sin{\left(x \right)}} \begin{cases} - \frac{i x^{4}}{2 y^{3} \left(\frac{x^{2}}{y^{2}} - 1\right)^{\frac{3}{2}}} + \frac{3 i x^{2}}{2 y \sqrt{\frac{x^{2}}{y^{2}} - 1}} + \frac{i x^{2}}{2 y \left(\frac{x^{2}}{y^{2}} - 1\right)^{\frac{3}{2}}} - \frac{i y}{\sqrt{\frac{x^{2}}{y^{2}} - 1}} & \text{for}\: \left|{\frac{x^{2}}{y^{2}}}\right| > 1 \\- \frac{x^{2}}{2 y \sqrt{- \frac{x^{2}}{y^{2}} + 1}} + \frac{y \sqrt{- \frac{x^{2}}{y^{2}} + 1}}{2} + \frac{y}{2 \sqrt{- \frac{x^{2}}{y^{2}} + 1}} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases}$$
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/ 0 | / | | | | / 2 4 2 | 2| | | | I*y I*x I*x 3*I*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 sin(x) < 0 | | | / x | | | y* / 1 - -- | | | / 2 2 | | | \/ y y x | | | ---------------- + ---------------- - ------------------ otherwise | | | 2 ________ ________ | | | / 2 / 2 | | | / x / x | | | 2* / 1 - -- 2*y* / 1 - -- | | | / 2 / 2 | | \ \/ y \/ y | | | / | sin(x) < | sin(x) | / | | | | / 2 4 2 | 2| | | | I*y I*x I*x 3*I*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 | | | y* / 1 - -- | | | / 2 2 | | | \/ y y x | | | ---------------- + ---------------- - ------------------ otherwise | | | 2 ________ ________ | | | / 2 / 2 | | | / x / x | | | 2* / 1 - -- 2*y* / 1 - -- | | | / 2 / 2 | | \ \/ y \/ y | | | / \ 0
$$\begin{cases} - \int\limits_{\sin{\left(x \right)}}^{0} \begin{cases} - \frac{i x^{4}}{2 y^{3} \left(\frac{x^{2}}{y^{2}} - 1\right)^{\frac{3}{2}}} + \frac{3 i x^{2}}{2 y \sqrt{\frac{x^{2}}{y^{2}} - 1}} + \frac{i x^{2}}{2 y \left(\frac{x^{2}}{y^{2}} - 1\right)^{\frac{3}{2}}} - \frac{i y}{\sqrt{\frac{x^{2}}{y^{2}} - 1}} & \text{for}\: \left|{\frac{x^{2}}{y^{2}}}\right| > 1 \\- \frac{x^{2}}{2 y \sqrt{- \frac{x^{2}}{y^{2}} + 1}} + \frac{y \sqrt{- \frac{x^{2}}{y^{2}} + 1}}{2} + \frac{y}{2 \sqrt{- \frac{x^{2}}{y^{2}} + 1}} & \text{otherwise} \end{cases}\, dx & \text{for}\: \sin{\left(x \right)} < 0 \\\int\limits_{0}^{\sin{\left(x \right)}} \begin{cases} - \frac{i x^{4}}{2 y^{3} \left(\frac{x^{2}}{y^{2}} - 1\right)^{\frac{3}{2}}} + \frac{3 i x^{2}}{2 y \sqrt{\frac{x^{2}}{y^{2}} - 1}} + \frac{i x^{2}}{2 y \left(\frac{x^{2}}{y^{2}} - 1\right)^{\frac{3}{2}}} - \frac{i y}{\sqrt{\frac{x^{2}}{y^{2}} - 1}} & \text{for}\: \left|{\frac{x^{2}}{y^{2}}}\right| > 1 \\- \frac{x^{2}}{2 y \sqrt{- \frac{x^{2}}{y^{2}} + 1}} + \frac{y \sqrt{- \frac{x^{2}}{y^{2}} + 1}}{2} + \frac{y}{2 \sqrt{- \frac{x^{2}}{y^{2}} + 1}} & \text{otherwise} \end{cases}\, dx & \text{otherwise} \end{cases}$$
Piecewise((-Integral(Piecewise((-i*y/sqrt(-1 + x^2/y^2) + i*x^2/(2*y*(-1 + x^2/y^2)^(3/2)) - i*x^4/(2*y^3*(-1 + x^2/y^2)^(3/2)) + 3*i*x^2/(2*y*sqrt(-1 + x^2/y^2)), |x^2/y^2| > 1), (y*sqrt(1 - x^2/y^2)/2 + y/(2*sqrt(1 - x^2/y^2)) - x^2/(2*y*sqrt(1 - x^2/y^2)), True)), (x, sin(x), 0)), sin(x) < 0), (Integral(Piecewise((-i*y/sqrt(-1 + x^2/y^2) + i*x^2/(2*y*(-1 + x^2/y^2)^(3/2)) - i*x^4/(2*y^3*(-1 + x^2/y^2)^(3/2)) + 3*i*x^2/(2*y*sqrt(-1 + x^2/y^2)), |x^2/y^2| > 1), (y*sqrt(1 - x^2/y^2)/2 + y/(2*sqrt(1 - x^2/y^2)) - x^2/(2*y*sqrt(1 - x^2/y^2)), True)), (x, 0, sin(x))), True))
Use the examples entering the upper and lower limits of integration.