Sr Examen
Integral de 1/(a^2+x^2)^5 dx
Solución
Ha introducido
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oo / | | 1 | ---------- dx | 5 | / 2 2\ | \a + x / | / -oo
$$\int\limits_{-\infty}^{\infty} \frac{1}{\left(a^{2} + x^{2}\right)^{5}}\, dx$$
Integral(1/((a^2 + x^2)^5), (x, -oo, oo))
Respuesta (Indefinida)
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/ 35*I*log(x - I*a) 35*I*log(x + I*a) | - ----------------- + ----------------- 7 6 2 5 4 3 | 1 256 256 105*x + 279*x*a + 385*a *x + 511*a *x | ---------- dx = C + --------------------------------------- + ------------------------------------------------------------- | 5 9 16 8 8 10 6 14 2 12 4 | / 2 2\ a 384*a + 384*a *x + 1536*a *x + 1536*a *x + 2304*a *x | \a + x / | /
$$\int \frac{1}{\left(a^{2} + x^{2}\right)^{5}}\, dx = C + \frac{279 a^{6} x + 511 a^{4} x^{3} + 385 a^{2} x^{5} + 105 x^{7}}{384 a^{16} + 1536 a^{14} x^{2} + 2304 a^{12} x^{4} + 1536 a^{10} x^{6} + 384 a^{8} x^{8}} + \frac{- \frac{35 i \log{\left(- i a + x \right)}}{256} + \frac{35 i \log{\left(i a + x \right)}}{256}}{a^{9}}$$
Respuesta
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/ 35*pi / / 1 \ \ | ------ for Or|And|2*|arg(a)| < pi, -- != 0, 2*|arg(a)| != pi|, 2*|arg(a)| < pi| | 9 | | 2 | | | 128*a \ \ a / / | | oo | / | | < | 1 | | ---------- dx otherwise | | 5 | | / 2 2\ | | \a + x / | | |/ |-oo \
$$\begin{cases} \frac{35 \pi}{128 a^{9}} & \text{for}\: \left(2 \left|{\arg{\left(a \right)}}\right| < \pi \wedge \frac{1}{a^{2}} \neq 0 \wedge 2 \left|{\arg{\left(a \right)}}\right| \neq \pi\right) \vee 2 \left|{\arg{\left(a \right)}}\right| < \pi \\\int\limits_{-\infty}^{\infty} \frac{1}{\left(a^{2} + x^{2}\right)^{5}}\, dx & \text{otherwise} \end{cases}$$
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/ 35*pi / / 1 \ \ | ------ for Or|And|2*|arg(a)| < pi, -- != 0, 2*|arg(a)| != pi|, 2*|arg(a)| < pi| | 9 | | 2 | | | 128*a \ \ a / / | | oo | / | | < | 1 | | ---------- dx otherwise | | 5 | | / 2 2\ | | \a + x / | | |/ |-oo \
$$\begin{cases} \frac{35 \pi}{128 a^{9}} & \text{for}\: \left(2 \left|{\arg{\left(a \right)}}\right| < \pi \wedge \frac{1}{a^{2}} \neq 0 \wedge 2 \left|{\arg{\left(a \right)}}\right| \neq \pi\right) \vee 2 \left|{\arg{\left(a \right)}}\right| < \pi \\\int\limits_{-\infty}^{\infty} \frac{1}{\left(a^{2} + x^{2}\right)^{5}}\, dx & \text{otherwise} \end{cases}$$
Piecewise((35*pi/(128*a^9), (2*Abs(arg(a)) < pi)∨((Ne(a^(-2), 0))∧(2*Abs(arg(a)) < pi)∧(Ne(2*Abs(arg(a)), pi)))), (Integral((a^2 + x^2)^(-5), (x, -oo, oo)), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.