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- Improper integral
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How to use it?
- Integral of d{x}:
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Integral of e^-x
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Integral of e^x/x
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Integral of 1/sin²(x)
- Integral of sqrt(1+x)/sqrt(1-x)
- Derivative of:
- sqrt(r^2-x^2)
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Identical expressions
- sqrt(r^ two -x^ two)
- square root of (r squared minus x squared )
- square root of (r to the power of two minus x to the power of two)
- √(r^2-x^2)
- sqrt(r2-x2)
- sqrtr2-x2
- sqrt(r²-x²)
- sqrt(r to the power of 2-x to the power of 2)
- sqrtr^2-x^2
- sqrt(r^2-x^2)dx
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Similar expressions
- sqrt(r^2+x^2)
Integral of sqrt(r^2-x^2) dx
The solution
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[src]
1 / | | _________ | / 2 2 | \/ r - x dx | / 0
$$\int\limits_{0}^{1} \sqrt{r^{2} - x^{2}}\, dx$$
Integral(sqrt(r^2 - x^2), (x, 0, 1))
The answer (Indefinite)
[src]
// 2 /x\ \
|| I*r *acosh|-| 3 | 2| |
|| \r/ I*x I*r*x |x | |
||- ------------- + ------------------- - ----------------- for |--| > 1|
|| 2 _________ _________ | 2| |
/ || / 2 / 2 |r | |
| || / x / x |
| _________ || 2*r* / -1 + -- 2* / -1 + -- |
| / 2 2 || / 2 / 2 |
| \/ r - x dx = C + |< \/ r \/ r |
| || |
/ || ________ |
|| / 2 |
|| / x |
|| 2 /x\ r*x* / 1 - -- |
|| r *asin|-| / 2 |
|| \r/ \/ r |
|| ---------- + ------------------ otherwise |
\\ 2 2 /
$$\int \sqrt{r^{2} - x^{2}}\, dx = C + \begin{cases} - \frac{i r^{2} \operatorname{acosh}{\left(\frac{x}{r} \right)}}{2} - \frac{i r x}{2 \sqrt{-1 + \frac{x^{2}}{r^{2}}}} + \frac{i x^{3}}{2 r \sqrt{-1 + \frac{x^{2}}{r^{2}}}} & \text{for}\: \left|{\frac{x^{2}}{r^{2}}}\right| > 1 \\\frac{r^{2} \operatorname{asin}{\left(\frac{x}{r} \right)}}{2} + \frac{r x \sqrt{1 - \frac{x^{2}}{r^{2}}}}{2} & \text{otherwise} \end{cases}$$
Use the examples entering the upper and lower limits of integration.