Sr Examen
Integral de 1/(k-y-y^2+k*y) dy
Solución
Ha introducido
[src]
1 / | | 1 | ---------------- dy | 2 | k - y - y + k*y | / 0
$$\int\limits_{0}^{1} \frac{1}{k y + \left(- y^{2} + \left(k - y\right)\right)}\, dy$$
Integral(1/(k - y - y^2 + k*y), (y, 0, 1))
Respuesta (Indefinida)
[src]
/ | | 1 log(2 + 2*y) log(-2*y + 2*k) | ---------------- dy = C + ------------ - --------------- | 2 1 + k 1 + k | k - y - y + k*y | /
$$\int \frac{1}{k y + \left(- y^{2} + \left(k - y\right)\right)}\, dy = C - \frac{\log{\left(2 k - 2 y \right)}}{k + 1} + \frac{\log{\left(2 y + 2 \right)}}{k + 1}$$
Respuesta
[src]
/ 2 \ / 2 \ / 2 \ / 2 \
|1 k 1 k k | |3 1 k k k | |1 1 k k k | |3 k 1 k k |
log|- - - - --------- - ----- - ---------| log|- + --------- - - + ----- + ---------| log|- + --------- - - + ----- + ---------| log|- - - - --------- - ----- - ---------|
\2 2 2*(1 + k) 1 + k 2*(1 + k)/ \2 2*(1 + k) 2 1 + k 2*(1 + k)/ \2 2*(1 + k) 2 1 + k 2*(1 + k)/ \2 2 2*(1 + k) 1 + k 2*(1 + k)/
------------------------------------------ + ------------------------------------------ - ------------------------------------------ - ------------------------------------------
1 + k 1 + k 1 + k 1 + k
$$\frac{\log{\left(- \frac{k^{2}}{2 \left(k + 1\right)} - \frac{k}{2} - \frac{k}{k + 1} + \frac{1}{2} - \frac{1}{2 \left(k + 1\right)} \right)}}{k + 1} - \frac{\log{\left(- \frac{k^{2}}{2 \left(k + 1\right)} - \frac{k}{2} - \frac{k}{k + 1} + \frac{3}{2} - \frac{1}{2 \left(k + 1\right)} \right)}}{k + 1} - \frac{\log{\left(\frac{k^{2}}{2 \left(k + 1\right)} - \frac{k}{2} + \frac{k}{k + 1} + \frac{1}{2} + \frac{1}{2 \left(k + 1\right)} \right)}}{k + 1} + \frac{\log{\left(\frac{k^{2}}{2 \left(k + 1\right)} - \frac{k}{2} + \frac{k}{k + 1} + \frac{3}{2} + \frac{1}{2 \left(k + 1\right)} \right)}}{k + 1}$$
=
=
/ 2 \ / 2 \ / 2 \ / 2 \
|1 k 1 k k | |3 1 k k k | |1 1 k k k | |3 k 1 k k |
log|- - - - --------- - ----- - ---------| log|- + --------- - - + ----- + ---------| log|- + --------- - - + ----- + ---------| log|- - - - --------- - ----- - ---------|
\2 2 2*(1 + k) 1 + k 2*(1 + k)/ \2 2*(1 + k) 2 1 + k 2*(1 + k)/ \2 2*(1 + k) 2 1 + k 2*(1 + k)/ \2 2 2*(1 + k) 1 + k 2*(1 + k)/
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1 + k 1 + k 1 + k 1 + k
$$\frac{\log{\left(- \frac{k^{2}}{2 \left(k + 1\right)} - \frac{k}{2} - \frac{k}{k + 1} + \frac{1}{2} - \frac{1}{2 \left(k + 1\right)} \right)}}{k + 1} - \frac{\log{\left(- \frac{k^{2}}{2 \left(k + 1\right)} - \frac{k}{2} - \frac{k}{k + 1} + \frac{3}{2} - \frac{1}{2 \left(k + 1\right)} \right)}}{k + 1} - \frac{\log{\left(\frac{k^{2}}{2 \left(k + 1\right)} - \frac{k}{2} + \frac{k}{k + 1} + \frac{1}{2} + \frac{1}{2 \left(k + 1\right)} \right)}}{k + 1} + \frac{\log{\left(\frac{k^{2}}{2 \left(k + 1\right)} - \frac{k}{2} + \frac{k}{k + 1} + \frac{3}{2} + \frac{1}{2 \left(k + 1\right)} \right)}}{k + 1}$$
log(1/2 - k/2 - 1/(2*(1 + k)) - k/(1 + k) - k^2/(2*(1 + k)))/(1 + k) + log(3/2 + 1/(2*(1 + k)) - k/2 + k/(1 + k) + k^2/(2*(1 + k)))/(1 + k) - log(1/2 + 1/(2*(1 + k)) - k/2 + k/(1 + k) + k^2/(2*(1 + k)))/(1 + k) - log(3/2 - k/2 - 1/(2*(1 + k)) - k/(1 + k) - k^2/(2*(1 + k)))/(1 + k)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.