Sr Examen
Integral de y^x*3^(x*y)*log(3) dx
Solución
Ha introducido
[src]
1 / | | x x*y | y *3 *log(3) dx | / 0
$$\int\limits_{0}^{1} 3^{x y} y^{x} \log{\left(3 \right)}\, dx$$
Integral((y^x*3^(x*y))*log(3), (x, 0, 1))
Respuesta
[src]
/ y | log(3) y*3 *log(3) / / W(log(3)) W(log(3)) \ / W(log(3))\\ | - ----------------- + ----------------- for Or|And|y != ---------, ----------------------------- != -1|, And|y > -oo, y < oo, y != ---------|| | y*log(3) + log(y) y*log(3) + log(y) \ \ log(3) -log(log(3)) + log(W(log(3))) / \ log(3) // | | / W(log(3)) \ | |1 + -----------------------------|*W(log(3)) < \ -log(log(3)) + log(W(log(3)))/ | log(3) e *log(3) W(log(3)) |- --------------------------------------------- + ----------------------------------------------------- for ----------------------------- != -1 | / W(log(3)) \ / W(log(3)) \ -log(log(3)) + log(W(log(3))) | |1 + -----------------------------|*W(log(3)) |1 + -----------------------------|*W(log(3)) | \ -log(log(3)) + log(W(log(3)))/ \ -log(log(3)) + log(W(log(3)))/ | \ log(3) otherwise
$$\begin{cases} \frac{3^{y} y \log{\left(3 \right)}}{y \log{\left(3 \right)} + \log{\left(y \right)}} - \frac{\log{\left(3 \right)}}{y \log{\left(3 \right)} + \log{\left(y \right)}} & \text{for}\: \left(y \neq \frac{W\left(\log{\left(3 \right)}\right)}{\log{\left(3 \right)}} \wedge \frac{W\left(\log{\left(3 \right)}\right)}{\log{\left(W\left(\log{\left(3 \right)}\right) \right)} - \log{\left(\log{\left(3 \right)} \right)}} \neq -1\right) \vee \left(y > -\infty \wedge y < \infty \wedge y \neq \frac{W\left(\log{\left(3 \right)}\right)}{\log{\left(3 \right)}}\right) \\- \frac{\log{\left(3 \right)}}{\left(\frac{W\left(\log{\left(3 \right)}\right)}{\log{\left(W\left(\log{\left(3 \right)}\right) \right)} - \log{\left(\log{\left(3 \right)} \right)}} + 1\right) W\left(\log{\left(3 \right)}\right)} + \frac{e^{\left(\frac{W\left(\log{\left(3 \right)}\right)}{\log{\left(W\left(\log{\left(3 \right)}\right) \right)} - \log{\left(\log{\left(3 \right)} \right)}} + 1\right) W\left(\log{\left(3 \right)}\right)} \log{\left(3 \right)}}{\left(\frac{W\left(\log{\left(3 \right)}\right)}{\log{\left(W\left(\log{\left(3 \right)}\right) \right)} - \log{\left(\log{\left(3 \right)} \right)}} + 1\right) W\left(\log{\left(3 \right)}\right)} & \text{for}\: \frac{W\left(\log{\left(3 \right)}\right)}{\log{\left(W\left(\log{\left(3 \right)}\right) \right)} - \log{\left(\log{\left(3 \right)} \right)}} \neq -1 \\\log{\left(3 \right)} & \text{otherwise} \end{cases}$$
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/ y | log(3) y*3 *log(3) / / W(log(3)) W(log(3)) \ / W(log(3))\\ | - ----------------- + ----------------- for Or|And|y != ---------, ----------------------------- != -1|, And|y > -oo, y < oo, y != ---------|| | y*log(3) + log(y) y*log(3) + log(y) \ \ log(3) -log(log(3)) + log(W(log(3))) / \ log(3) // | | / W(log(3)) \ | |1 + -----------------------------|*W(log(3)) < \ -log(log(3)) + log(W(log(3)))/ | log(3) e *log(3) W(log(3)) |- --------------------------------------------- + ----------------------------------------------------- for ----------------------------- != -1 | / W(log(3)) \ / W(log(3)) \ -log(log(3)) + log(W(log(3))) | |1 + -----------------------------|*W(log(3)) |1 + -----------------------------|*W(log(3)) | \ -log(log(3)) + log(W(log(3)))/ \ -log(log(3)) + log(W(log(3)))/ | \ log(3) otherwise
$$\begin{cases} \frac{3^{y} y \log{\left(3 \right)}}{y \log{\left(3 \right)} + \log{\left(y \right)}} - \frac{\log{\left(3 \right)}}{y \log{\left(3 \right)} + \log{\left(y \right)}} & \text{for}\: \left(y \neq \frac{W\left(\log{\left(3 \right)}\right)}{\log{\left(3 \right)}} \wedge \frac{W\left(\log{\left(3 \right)}\right)}{\log{\left(W\left(\log{\left(3 \right)}\right) \right)} - \log{\left(\log{\left(3 \right)} \right)}} \neq -1\right) \vee \left(y > -\infty \wedge y < \infty \wedge y \neq \frac{W\left(\log{\left(3 \right)}\right)}{\log{\left(3 \right)}}\right) \\- \frac{\log{\left(3 \right)}}{\left(\frac{W\left(\log{\left(3 \right)}\right)}{\log{\left(W\left(\log{\left(3 \right)}\right) \right)} - \log{\left(\log{\left(3 \right)} \right)}} + 1\right) W\left(\log{\left(3 \right)}\right)} + \frac{e^{\left(\frac{W\left(\log{\left(3 \right)}\right)}{\log{\left(W\left(\log{\left(3 \right)}\right) \right)} - \log{\left(\log{\left(3 \right)} \right)}} + 1\right) W\left(\log{\left(3 \right)}\right)} \log{\left(3 \right)}}{\left(\frac{W\left(\log{\left(3 \right)}\right)}{\log{\left(W\left(\log{\left(3 \right)}\right) \right)} - \log{\left(\log{\left(3 \right)} \right)}} + 1\right) W\left(\log{\left(3 \right)}\right)} & \text{for}\: \frac{W\left(\log{\left(3 \right)}\right)}{\log{\left(W\left(\log{\left(3 \right)}\right) \right)} - \log{\left(\log{\left(3 \right)} \right)}} \neq -1 \\\log{\left(3 \right)} & \text{otherwise} \end{cases}$$
Piecewise((-log(3)/(y*log(3) + log(y)) + y*3^y*log(3)/(y*log(3) + log(y)), ((y > -oo)∧(y < oo)∧(Ne(y, LambertW(log(3))/log(3))))∨((Ne(y, LambertW(log(3))/log(3)))∧(Ne(LambertW(log(3))/(-log(log(3)) + log(LambertW(log(3)))), -1)))), (-log(3)/((1 + LambertW(log(3))/(-log(log(3)) + log(LambertW(log(3)))))*LambertW(log(3))) + exp((1 + LambertW(log(3))/(-log(log(3)) + log(LambertW(log(3)))))*LambertW(log(3)))*log(3)/((1 + LambertW(log(3))/(-log(log(3)) + log(LambertW(log(3)))))*LambertW(log(3))), Ne(LambertW(log(3))/(-log(log(3)) + log(LambertW(log(3)))), -1)), (log(3), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.