Sr Examen
Integral de sqrt(x)ln^2(x) dx
Solución
Ha introducido
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o / | | ___ 2 | \/ x *log (x) dx | / 0
$$\int\limits_{0}^{o} \sqrt{x} \log{\left(x \right)}^{2}\, dx$$
Integral(sqrt(x)*log(x)^2, (x, 0, o))
Respuesta (Indefinida)
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/ | 3/2 3/2 3/2 2 | ___ 2 16*x 8*x *log(x) 2*x *log (x) | \/ x *log (x) dx = C + ------- - ------------- + -------------- | 27 9 3 /
$$\int \sqrt{x} \log{\left(x \right)}^{2}\, dx = C + \frac{2 x^{\frac{3}{2}} \log{\left(x \right)}^{2}}{3} - \frac{8 x^{\frac{3}{2}} \log{\left(x \right)}}{9} + \frac{16 x^{\frac{3}{2}}}{27}$$
Respuesta
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/ 3/2 / 1 \ 3/2 2/ 1 \ 3/2 /1\ 3/2 2/1\ | 3/2 8*Max(-1, o) *log|----------| 3/2 2*Max(-1, o) *log |----------| 3/2 2 8*I*(-o) *log|-| 2*I*(-o) *log |-| | 16*I*(-o) \Max(-1, o)/ 8*Max(-1, o) *log(Max(-1, o)) \Max(-1, o)/ 2*Max(-1, o) *log (Max(-1, o)) \o/ \o/ <- ------------ - ------------------------------- - ------------------------------- - -------------------------------- + -------------------------------- - ------------------ - ------------------- for o < 0 | 27 9 9 3 3 9 3 | \ nan otherwise
$$\begin{cases} - \frac{2 i \left(- o\right)^{\frac{3}{2}} \log{\left(\frac{1}{o} \right)}^{2}}{3} - \frac{8 i \left(- o\right)^{\frac{3}{2}} \log{\left(\frac{1}{o} \right)}}{9} - \frac{16 i \left(- o\right)^{\frac{3}{2}}}{27} - \frac{2 \log{\left(\frac{1}{\max\left(-1, o\right)} \right)}^{2} \max\left(-1, o\right)^{\frac{3}{2}}}{3} - \frac{8 \log{\left(\frac{1}{\max\left(-1, o\right)} \right)} \max\left(-1, o\right)^{\frac{3}{2}}}{9} + \frac{2 \log{\left(\max\left(-1, o\right) \right)}^{2} \max\left(-1, o\right)^{\frac{3}{2}}}{3} - \frac{8 \log{\left(\max\left(-1, o\right) \right)} \max\left(-1, o\right)^{\frac{3}{2}}}{9} & \text{for}\: o < 0 \\\text{NaN} & \text{otherwise} \end{cases}$$
=
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/ 3/2 / 1 \ 3/2 2/ 1 \ 3/2 /1\ 3/2 2/1\ | 3/2 8*Max(-1, o) *log|----------| 3/2 2*Max(-1, o) *log |----------| 3/2 2 8*I*(-o) *log|-| 2*I*(-o) *log |-| | 16*I*(-o) \Max(-1, o)/ 8*Max(-1, o) *log(Max(-1, o)) \Max(-1, o)/ 2*Max(-1, o) *log (Max(-1, o)) \o/ \o/ <- ------------ - ------------------------------- - ------------------------------- - -------------------------------- + -------------------------------- - ------------------ - ------------------- for o < 0 | 27 9 9 3 3 9 3 | \ nan otherwise
$$\begin{cases} - \frac{2 i \left(- o\right)^{\frac{3}{2}} \log{\left(\frac{1}{o} \right)}^{2}}{3} - \frac{8 i \left(- o\right)^{\frac{3}{2}} \log{\left(\frac{1}{o} \right)}}{9} - \frac{16 i \left(- o\right)^{\frac{3}{2}}}{27} - \frac{2 \log{\left(\frac{1}{\max\left(-1, o\right)} \right)}^{2} \max\left(-1, o\right)^{\frac{3}{2}}}{3} - \frac{8 \log{\left(\frac{1}{\max\left(-1, o\right)} \right)} \max\left(-1, o\right)^{\frac{3}{2}}}{9} + \frac{2 \log{\left(\max\left(-1, o\right) \right)}^{2} \max\left(-1, o\right)^{\frac{3}{2}}}{3} - \frac{8 \log{\left(\max\left(-1, o\right) \right)} \max\left(-1, o\right)^{\frac{3}{2}}}{9} & \text{for}\: o < 0 \\\text{NaN} & \text{otherwise} \end{cases}$$
Piecewise((-16*i*(-o)^(3/2)/27 - 8*Max(-1, o)^(3/2)*log(1/Max(-1, o))/9 - 8*Max(-1, o)^(3/2)*log(Max(-1, o))/9 - 2*Max(-1, o)^(3/2)*log(1/Max(-1, o))^2/3 + 2*Max(-1, o)^(3/2)*log(Max(-1, o))^2/3 - 8*i*(-o)^(3/2)*log(1/o)/9 - 2*i*(-o)^(3/2)*log(1/o)^2/3, o < 0), (nan, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.