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- Improper integral
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How to use it?
- Integral of d{x}:
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Integral of √xdx
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Integral of √(1-x^2)
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Integral of 1/(2x)
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Integral of (e^x)/x
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Identical expressions
- e^(-(x^(one /(z+ one))))
- e to the power of ( minus (x to the power of (1 divide by (z plus 1))))
- e to the power of ( minus (x to the power of (one divide by (z plus one))))
- e(-(x(1/(z+1))))
- e-x1/z+1
- e^-x^1/z+1
- e^(-(x^(1 divide by (z+1))))
- e^(-(x^(1/(z+1))))dx
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Similar expressions
- e^(-(x^(1/(z-1))))
- e^((x^(1/(z+1))))
Integral of e^(-(x^(1/(z+1)))) dx
The solution
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[src]
1 / | | 1 | ----- | z + 1 | -x | E dx | / 0
$$\int\limits_{0}^{1} e^{- x^{\frac{1}{z + 1}}}\, dx$$
Integral(E^(-x^(1/(z + 1))), (x, 0, 1))
The answer (Indefinite)
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/ | | 1 / 1 \ / 1 \ / 1 \ | ----- | -----| | -----| | -----| | z + 1 | 1 + z| 2 | 1 + z| | 1 + z| | -x Gamma(1 + z)*lowergamma\1 + z, x / z *Gamma(1 + z)*lowergamma\1 + z, x / 2*z*Gamma(1 + z)*lowergamma\1 + z, x / | E dx = C + -------------------------------------- + ----------------------------------------- + ------------------------------------------ | Gamma(2 + z) Gamma(2 + z) Gamma(2 + z) /
$$\int e^{- x^{\frac{1}{z + 1}}}\, dx = C + \frac{z^{2} \Gamma\left(z + 1\right) \gamma\left(z + 1, x^{\frac{1}{z + 1}}\right)}{\Gamma\left(z + 2\right)} + \frac{2 z \Gamma\left(z + 1\right) \gamma\left(z + 1, x^{\frac{1}{z + 1}}\right)}{\Gamma\left(z + 2\right)} + \frac{\Gamma\left(z + 1\right) \gamma\left(z + 1, x^{\frac{1}{z + 1}}\right)}{\Gamma\left(z + 2\right)}$$
The answer
[src]
/ 1 \ / 1 \ / 1 \
| -----| | -----| | -----|
| 1 + z| 2 2 | 1 + z| | 1 + z|
Gamma(1 + z)*lowergamma(1 + z, 1) Gamma(1 + z)*lowergamma\1 + z, 0 / z *Gamma(1 + z)*lowergamma(1 + z, 1) z *Gamma(1 + z)*lowergamma\1 + z, 0 / 2*z*Gamma(1 + z)*lowergamma\1 + z, 0 / 2*z*Gamma(1 + z)*lowergamma(1 + z, 1)
--------------------------------- - -------------------------------------- + ------------------------------------ - ----------------------------------------- - ------------------------------------------ + -------------------------------------
Gamma(2 + z) Gamma(2 + z) Gamma(2 + z) Gamma(2 + z) Gamma(2 + z) Gamma(2 + z)
$$\frac{z^{2} \Gamma\left(z + 1\right) \gamma\left(z + 1, 1\right)}{\Gamma\left(z + 2\right)} - \frac{z^{2} \Gamma\left(z + 1\right) \gamma\left(z + 1, 0^{\frac{1}{z + 1}}\right)}{\Gamma\left(z + 2\right)} + \frac{2 z \Gamma\left(z + 1\right) \gamma\left(z + 1, 1\right)}{\Gamma\left(z + 2\right)} - \frac{2 z \Gamma\left(z + 1\right) \gamma\left(z + 1, 0^{\frac{1}{z + 1}}\right)}{\Gamma\left(z + 2\right)} + \frac{\Gamma\left(z + 1\right) \gamma\left(z + 1, 1\right)}{\Gamma\left(z + 2\right)} - \frac{\Gamma\left(z + 1\right) \gamma\left(z + 1, 0^{\frac{1}{z + 1}}\right)}{\Gamma\left(z + 2\right)}$$
=
=
/ 1 \ / 1 \ / 1 \
| -----| | -----| | -----|
| 1 + z| 2 2 | 1 + z| | 1 + z|
Gamma(1 + z)*lowergamma(1 + z, 1) Gamma(1 + z)*lowergamma\1 + z, 0 / z *Gamma(1 + z)*lowergamma(1 + z, 1) z *Gamma(1 + z)*lowergamma\1 + z, 0 / 2*z*Gamma(1 + z)*lowergamma\1 + z, 0 / 2*z*Gamma(1 + z)*lowergamma(1 + z, 1)
--------------------------------- - -------------------------------------- + ------------------------------------ - ----------------------------------------- - ------------------------------------------ + -------------------------------------
Gamma(2 + z) Gamma(2 + z) Gamma(2 + z) Gamma(2 + z) Gamma(2 + z) Gamma(2 + z)
$$\frac{z^{2} \Gamma\left(z + 1\right) \gamma\left(z + 1, 1\right)}{\Gamma\left(z + 2\right)} - \frac{z^{2} \Gamma\left(z + 1\right) \gamma\left(z + 1, 0^{\frac{1}{z + 1}}\right)}{\Gamma\left(z + 2\right)} + \frac{2 z \Gamma\left(z + 1\right) \gamma\left(z + 1, 1\right)}{\Gamma\left(z + 2\right)} - \frac{2 z \Gamma\left(z + 1\right) \gamma\left(z + 1, 0^{\frac{1}{z + 1}}\right)}{\Gamma\left(z + 2\right)} + \frac{\Gamma\left(z + 1\right) \gamma\left(z + 1, 1\right)}{\Gamma\left(z + 2\right)} - \frac{\Gamma\left(z + 1\right) \gamma\left(z + 1, 0^{\frac{1}{z + 1}}\right)}{\Gamma\left(z + 2\right)}$$
gamma(1 + z)*lowergamma(1 + z, 1)/gamma(2 + z) - gamma(1 + z)*lowergamma(1 + z, 0^(1/(1 + z)))/gamma(2 + z) + z^2*gamma(1 + z)*lowergamma(1 + z, 1)/gamma(2 + z) - z^2*gamma(1 + z)*lowergamma(1 + z, 0^(1/(1 + z)))/gamma(2 + z) - 2*z*gamma(1 + z)*lowergamma(1 + z, 0^(1/(1 + z)))/gamma(2 + z) + 2*z*gamma(1 + z)*lowergamma(1 + z, 1)/gamma(2 + z)
Use the examples entering the upper and lower limits of integration.