Sr Examen
Integral de (x-1)/(t^22*x-1) dx
Solución
Ha introducido
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1 / | | x - 1 | --------- dx | 22 | t *x - 1 | / 0
$$\int\limits_{0}^{1} \frac{x - 1}{t^{22} x - 1}\, dx$$
Integral((x - 1)/(t^22*x - 1), (x, 0, 1))
Respuesta (Indefinida)
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// 22 \
|| -x for t = 0|
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/ 2 3 4 5 6 7 8 9 10\ / 2 4 6 8 10 3 5 7 9\ || / 22\ |
(1 + t)*(-1 + t)*\1 + t + t + t + t + t + t + t + t + t + t /*\1 + t + t + t + t + t - t - t - t - t - t /*|
$$\int \frac{x - 1}{t^{22} x - 1}\, dx = C + \frac{x}{t^{22}} - \frac{\left(t - 1\right) \left(t + 1\right) \left(t^{10} - t^{9} + t^{8} - t^{7} + t^{6} - t^{5} + t^{4} - t^{3} + t^{2} - t + 1\right) \left(t^{10} + t^{9} + t^{8} + t^{7} + t^{6} + t^{5} + t^{4} + t^{3} + t^{2} + t + 1\right) \left(\begin{cases} - x & \text{for}\: t^{22} = 0 \\\frac{\log{\left(t^{22} x - 1 \right)}}{t^{22}} & \text{otherwise} \end{cases}\right)}{t^{22}}$$
Respuesta
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/ 2 3 4 5 6 7 8 9 10\ / 2 4 6 8 10 3 5 7 9\ / 22\ / 2 3 4 5 6 7 8 9 10\ / 2 4 6 8 10 3 5 7 9\ 1 (1 + t)*(-1 + t)*\1 + t + t + t + t + t + t + t + t + t + t /*\1 + t + t + t + t + t - t - t - t - t - t /*log\-1 + t / pi*I*(1 + t)*(-1 + t)*\1 + t + t + t + t + t + t + t + t + t + t /*\1 + t + t + t + t + t - t - t - t - t - t / --- - ------------------------------------------------------------------------------------------------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------------------- 22 44 44 t t t
$$\frac{1}{t^{22}} - \frac{\left(t - 1\right) \left(t + 1\right) \left(t^{10} - t^{9} + t^{8} - t^{7} + t^{6} - t^{5} + t^{4} - t^{3} + t^{2} - t + 1\right) \left(t^{10} + t^{9} + t^{8} + t^{7} + t^{6} + t^{5} + t^{4} + t^{3} + t^{2} + t + 1\right) \log{\left(t^{22} - 1 \right)}}{t^{44}} + \frac{i \pi \left(t - 1\right) \left(t + 1\right) \left(t^{10} - t^{9} + t^{8} - t^{7} + t^{6} - t^{5} + t^{4} - t^{3} + t^{2} - t + 1\right) \left(t^{10} + t^{9} + t^{8} + t^{7} + t^{6} + t^{5} + t^{4} + t^{3} + t^{2} + t + 1\right)}{t^{44}}$$
=
=
/ 2 3 4 5 6 7 8 9 10\ / 2 4 6 8 10 3 5 7 9\ / 22\ / 2 3 4 5 6 7 8 9 10\ / 2 4 6 8 10 3 5 7 9\ 1 (1 + t)*(-1 + t)*\1 + t + t + t + t + t + t + t + t + t + t /*\1 + t + t + t + t + t - t - t - t - t - t /*log\-1 + t / pi*I*(1 + t)*(-1 + t)*\1 + t + t + t + t + t + t + t + t + t + t /*\1 + t + t + t + t + t - t - t - t - t - t / --- - ------------------------------------------------------------------------------------------------------------------------------------------ + --------------------------------------------------------------------------------------------------------------------------------- 22 44 44 t t t
$$\frac{1}{t^{22}} - \frac{\left(t - 1\right) \left(t + 1\right) \left(t^{10} - t^{9} + t^{8} - t^{7} + t^{6} - t^{5} + t^{4} - t^{3} + t^{2} - t + 1\right) \left(t^{10} + t^{9} + t^{8} + t^{7} + t^{6} + t^{5} + t^{4} + t^{3} + t^{2} + t + 1\right) \log{\left(t^{22} - 1 \right)}}{t^{44}} + \frac{i \pi \left(t - 1\right) \left(t + 1\right) \left(t^{10} - t^{9} + t^{8} - t^{7} + t^{6} - t^{5} + t^{4} - t^{3} + t^{2} - t + 1\right) \left(t^{10} + t^{9} + t^{8} + t^{7} + t^{6} + t^{5} + t^{4} + t^{3} + t^{2} + t + 1\right)}{t^{44}}$$
t^(-22) - (1 + t)*(-1 + t)*(1 + t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7 + t^8 + t^9 + t^10)*(1 + t^2 + t^4 + t^6 + t^8 + t^10 - t - t^3 - t^5 - t^7 - t^9)*log(-1 + t^22)/t^44 + pi*i*(1 + t)*(-1 + t)*(1 + t + t^2 + t^3 + t^4 + t^5 + t^6 + t^7 + t^8 + t^9 + t^10)*(1 + t^2 + t^4 + t^6 + t^8 + t^10 - t - t^3 - t^5 - t^7 - t^9)/t^44
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.