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- Improper integral
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How to use it?
- Integral of d{x}:
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Integral of x^3/(x-1)
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Integral of x*2^x
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Integral of sin^5
- Integral of x^2*a^x
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Identical expressions
- x^ two *a^x
- x squared multiply by a to the power of x
- x to the power of two multiply by a to the power of x
- x2*ax
- x²*a^x
- x to the power of 2*a to the power of x
- x^2a^x
- x2ax
- x^2*a^xdx
Integral of x^2*a^x dx
The solution
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1 / | | 2 x | x *a dx | / 0
$$\int\limits_{0}^{1} a^{x} x^{2}\, dx$$
Integral(x^2*a^x, (x, 0, 1))
The answer (Indefinite)
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// x / 2 2 \ \
||a *\2 + x *log (a) - 2*x*log(a)/ 3 |
/ ||-------------------------------- for log (a) != 0|
| || 3 |
| 2 x || log (a) |
| x *a dx = C + |< |
| || 3 |
/ || x |
|| -- otherwise |
|| 3 |
\\ /
$$\int a^{x} x^{2}\, dx = C + \begin{cases} \frac{a^{x} \left(x^{2} \log{\left(a \right)}^{2} - 2 x \log{\left(a \right)} + 2\right)}{\log{\left(a \right)}^{3}} & \text{for}\: \log{\left(a \right)}^{3} \neq 0 \\\frac{x^{3}}{3} & \text{otherwise} \end{cases}$$
The answer
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/ / 2 \ | 2 a*\2 + log (a) - 2*log(a)/ |- ------- + -------------------------- for Or(And(a >= 0, a < 1), a > 1) < 3 3 | log (a) log (a) | \ 1/3 otherwise
$$\begin{cases} \frac{a \left(\log{\left(a \right)}^{2} - 2 \log{\left(a \right)} + 2\right)}{\log{\left(a \right)}^{3}} - \frac{2}{\log{\left(a \right)}^{3}} & \text{for}\: \left(a \geq 0 \wedge a < 1\right) \vee a > 1 \\\frac{1}{3} & \text{otherwise} \end{cases}$$
=
=
/ / 2 \ | 2 a*\2 + log (a) - 2*log(a)/ |- ------- + -------------------------- for Or(And(a >= 0, a < 1), a > 1) < 3 3 | log (a) log (a) | \ 1/3 otherwise
$$\begin{cases} \frac{a \left(\log{\left(a \right)}^{2} - 2 \log{\left(a \right)} + 2\right)}{\log{\left(a \right)}^{3}} - \frac{2}{\log{\left(a \right)}^{3}} & \text{for}\: \left(a \geq 0 \wedge a < 1\right) \vee a > 1 \\\frac{1}{3} & \text{otherwise} \end{cases}$$
Piecewise((-2/log(a)^3 + a*(2 + log(a)^2 - 2*log(a))/log(a)^3, (a > 1)∨((a >= 0)∧(a < 1))), (1/3, True))
Use the examples entering the upper and lower limits of integration.