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- Improper integral
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How to use it?
- Integral of d{x}:
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Integral of 1-7*x^2
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Integral of (x+1)dx
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Integral of (x^2+1)/((x^3+3x+1)^5)
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Integral of dx/(e^x-1)
- Derivative of:
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x^2*sqrt(1-x^2)
- Graphing y =:
- x^2*sqrt(1-x^2)
- Canonical form:
- x^2*sqrt(1-x^2)
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Identical expressions
- x^ two *sqrt(one -x^ two)
- x squared multiply by square root of (1 minus x squared )
- x to the power of two multiply by square root of (one minus x to the power of two)
- x^2*√(1-x^2)
- x2*sqrt(1-x2)
- x2*sqrt1-x2
- x²*sqrt(1-x²)
- x to the power of 2*sqrt(1-x to the power of 2)
- x^2sqrt(1-x^2)
- x2sqrt(1-x2)
- x2sqrt1-x2
- x^2sqrt1-x^2
- x^2*sqrt(1-x^2)dx
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Similar expressions
- x^2*sqrt(1+x^2)
- ((x^2)-(3*pi/32))(sqrt(1-x^2))*(sqrt(1-x^2))
Integral of x^2*sqrt(1-x^2) dx
The solution
You have entered
[src]
1 / | | ________ | 2 / 2 | x *\/ 1 - x dx | / 0
$$\int\limits_{0}^{1} x^{2} \sqrt{- x^{2} + 1}\, dx$$
Integral(x^2*sqrt(1 - x^2), (x, 0, 1))
Detail solution
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Rewrite the integrand:
SqrtQuadraticDenomRule(a=1, b=0, c=-1, coeffs=[-1, 0, 1, 0, 0], context=(-x**4 + x**2)/sqrt(1 - x**2), symbol=x)
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Now simplify:
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Add the constant of integration:
The answer is:
The answer (Indefinite)
[src]
/ | | ________ ________ / 3\ | 2 / 2 asin(x) / 2 | x x | | x *\/ 1 - x dx = C + ------- + \/ 1 - x *|- - + --| | 8 \ 8 4 / /
$${{\arcsin x}\over{8}}-{{x\,\left(1-x^2\right)^{{{3}\over{2}}}
}\over{4}}+{{x\,\sqrt{1-x^2}}\over{8}}$$
The graph
The answer
[src]
1 / | | / 2 6 2 4 4 | | I 9*I*x I*x I*x I 3*I*x 5*I*x 2 | |-------------- - -------------- - -------------- - -------------- - ---------------------- + -------------- + -------------- for x > 1 | | _________ _________ 3/2 3/2 _______ ________ 3/2 _________ | | / 2 / 2 / 2\ / 2\ 8*\/ 1 + x *\/ -1 + x / 2\ / 2 | |8*\/ -1 + x 8*\/ -1 + x 4*\-1 + x / 8*\-1 + x / 8*\-1 + x / 4*\/ -1 + x | < dx | | 4 6 2 4 2 | | 5*x x x 3*x 9*x | | - ------------- - ------------- - ------------- + ------------- + ------------- otherwise | | ________ 3/2 3/2 3/2 ________ | | / 2 / 2\ / 2\ / 2\ / 2 | \ 4*\/ 1 - x 4*\1 - x / 8*\1 - x / 8*\1 - x / 8*\/ 1 - x | / 0
$${{\pi}\over{16}}$$
=
=
1 / | | / 2 6 2 4 4 | | I 9*I*x I*x I*x I 3*I*x 5*I*x 2 | |-------------- - -------------- - -------------- - -------------- - ---------------------- + -------------- + -------------- for x > 1 | | _________ _________ 3/2 3/2 _______ ________ 3/2 _________ | | / 2 / 2 / 2\ / 2\ 8*\/ 1 + x *\/ -1 + x / 2\ / 2 | |8*\/ -1 + x 8*\/ -1 + x 4*\-1 + x / 8*\-1 + x / 8*\-1 + x / 4*\/ -1 + x | < dx | | 4 6 2 4 2 | | 5*x x x 3*x 9*x | | - ------------- - ------------- - ------------- + ------------- + ------------- otherwise | | ________ 3/2 3/2 3/2 ________ | | / 2 / 2\ / 2\ / 2\ / 2 | \ 4*\/ 1 - x 4*\1 - x / 8*\1 - x / 8*\1 - x / 8*\/ 1 - x | / 0
$$\int\limits_{0}^{1} \begin{cases} - \frac{i x^{6}}{4 \left(x^{2} - 1\right)^{\frac{3}{2}}} + \frac{5 i x^{4}}{4 \sqrt{x^{2} - 1}} + \frac{3 i x^{4}}{8 \left(x^{2} - 1\right)^{\frac{3}{2}}} - \frac{9 i x^{2}}{8 \sqrt{x^{2} - 1}} - \frac{i x^{2}}{8 \left(x^{2} - 1\right)^{\frac{3}{2}}} + \frac{i}{8 \sqrt{x^{2} - 1}} - \frac{i}{8 \sqrt{x - 1} \sqrt{x + 1}} & \text{for}\: x^{2} > 1 \\- \frac{x^{6}}{4 \left(- x^{2} + 1\right)^{\frac{3}{2}}} - \frac{5 x^{4}}{4 \sqrt{- x^{2} + 1}} + \frac{3 x^{4}}{8 \left(- x^{2} + 1\right)^{\frac{3}{2}}} + \frac{9 x^{2}}{8 \sqrt{- x^{2} + 1}} - \frac{x^{2}}{8 \left(- x^{2} + 1\right)^{\frac{3}{2}}} & \text{otherwise} \end{cases}\, dx$$
The graph
Use the examples entering the upper and lower limits of integration.