Sr Examen
Integral de (a*x^2+b*x+c)/(x^3-8) dx
Solución
Ha introducido
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1 / | | 2 | a*x + b*x + c | -------------- dx | 3 | x - 8 | / 0
$$\int\limits_{0}^{1} \frac{c + \left(a x^{2} + b x\right)}{x^{3} - 8}\, dx$$
Integral((a*x^2 + b*x + c)/(x^3 - 8), (x, 0, 1))
Respuesta (Indefinida)
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/ / ___ \ | / c\ / 2 \ ___ |\/ 3 *(1 + x)| | 2 |-b + 4*a - -|*log\4 + x + 2*x/ \/ 3 *(-3*c + 6*b)*atan|-------------| | a*x + b*x + c /a b c \ \ 2/ \ 3 / | -------------- dx = C + |- + - + --|*log(-2 + x) + -------------------------------- + -------------------------------------- | 3 \3 6 12/ 12 36 | x - 8 | /
$$\int \frac{c + \left(a x^{2} + b x\right)}{x^{3} - 8}\, dx = C + \frac{\sqrt{3} \left(6 b - 3 c\right) \operatorname{atan}{\left(\frac{\sqrt{3} \left(x + 1\right)}{3} \right)}}{36} + \left(\frac{a}{3} + \frac{b}{6} + \frac{c}{12}\right) \log{\left(x - 2 \right)} + \frac{\left(4 a - b - \frac{c}{2}\right) \log{\left(x^{2} + 2 x + 4 \right)}}{12}$$
Respuesta
[src]
/ 2 \ / 2 \ / 2 \ / 2 \ / 2 2 2 2 2 \ / 2 2 2 2 2 \
| / ___ \ / ___ \ / ___ \| | / ___ \ / ___ \ / ___ \| | / ___ \ / ___ \ / ___ \| | / ___ \ / ___ \ / ___ \| |- 64*b*a - 8*a*c - 4*b*(c + 2*b + 4*a) + 2*c *(c + 2*b + 4*a) + 16*c*b + 32*a*b*(c + 2*b + 4*a)| | - 64*b*a - 8*a*c - 4*b*(c + 2*b + 4*a) + 2*c *(c + 2*b + 4*a) + 16*c*b + 32*a*b*(c + 2*b + 4*a)|
| | b c a I*\/ 3 *(-c + 2*b)| 2 2 2 2 | b c a I*\/ 3 *(-c + 2*b)| | b c a I*\/ 3 *(-c + 2*b)|| | | b c a I*\/ 3 *(-c + 2*b)| 2 2 2 2 | b c a I*\/ 3 *(-c + 2*b)| | b c a I*\/ 3 *(-c + 2*b)|| | | b c a I*\/ 3 *(-c + 2*b)| 2 2 2 2 | b c a I*\/ 3 *(-c + 2*b)| | b c a I*\/ 3 *(-c + 2*b)|| | | b c a I*\/ 3 *(-c + 2*b)| 2 2 2 2 | b c a I*\/ 3 *(-c + 2*b)| | b c a I*\/ 3 *(-c + 2*b)|| (c + 2*b + 4*a)*log|---------------------------------------------------------------------------------------------------| (c + 2*b + 4*a)*log|1 + ---------------------------------------------------------------------------------------------------|
/ ___ \ | - 576*b*|- -- - -- + - - ------------------| - 64*b*a - 8*a*c + 16*c*b + 24*c *|- -- - -- + - - ------------------| + 384*a*b*|- -- - -- + - - ------------------|| / ___ \ | - 576*b*|- -- - -- + - + ------------------| - 64*b*a - 8*a*c + 16*c*b + 24*c *|- -- - -- + - + ------------------| + 384*a*b*|- -- - -- + - + ------------------|| / ___ \ |- 576*b*|- -- - -- + - - ------------------| - 64*b*a - 8*a*c + 16*c*b + 24*c *|- -- - -- + - - ------------------| + 384*a*b*|- -- - -- + - - ------------------|| / ___ \ |- 576*b*|- -- - -- + - + ------------------| - 64*b*a - 8*a*c + 16*c*b + 24*c *|- -- - -- + - + ------------------| + 384*a*b*|- -- - -- + - + ------------------|| | 3 3 | | 3 3 |
| b c a I*\/ 3 *(-c + 2*b)| | \ 12 24 3 24 / \ 12 24 3 24 / \ 12 24 3 24 /| | b c a I*\/ 3 *(-c + 2*b)| | \ 12 24 3 24 / \ 12 24 3 24 / \ 12 24 3 24 /| | b c a I*\/ 3 *(-c + 2*b)| | \ 12 24 3 24 / \ 12 24 3 24 / \ 12 24 3 24 /| | b c a I*\/ 3 *(-c + 2*b)| | \ 12 24 3 24 / \ 12 24 3 24 / \ 12 24 3 24 /| \ - c + 8*b / \ - c + 8*b /
|- -- - -- + - - ------------------|*log|1 + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------| + |- -- - -- + - + ------------------|*log|1 + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------| - |- -- - -- + - - ------------------|*log|----------------------------------------------------------------------------------------------------------------------------------------------------------------------| - |- -- - -- + - + ------------------|*log|----------------------------------------------------------------------------------------------------------------------------------------------------------------------| - ------------------------------------------------------------------------------------------------------------------------ + ----------------------------------------------------------------------------------------------------------------------------
\ 12 24 3 24 / | 3 3 | \ 12 24 3 24 / | 3 3 | \ 12 24 3 24 / | 3 3 | \ 12 24 3 24 / | 3 3 | 12 12
\ - c + 8*b / \ - c + 8*b / \ - c + 8*b / \ - c + 8*b /
$$- \frac{\left(4 a + 2 b + c\right) \log{\left(\frac{- 64 a^{2} b + 32 a b \left(4 a + 2 b + c\right) - 8 a c^{2} + 16 b^{2} c - 4 b \left(4 a + 2 b + c\right)^{2} + 2 c^{2} \left(4 a + 2 b + c\right)}{8 b^{3} - c^{3}} \right)}}{12} + \frac{\left(4 a + 2 b + c\right) \log{\left(1 + \frac{- 64 a^{2} b + 32 a b \left(4 a + 2 b + c\right) - 8 a c^{2} + 16 b^{2} c - 4 b \left(4 a + 2 b + c\right)^{2} + 2 c^{2} \left(4 a + 2 b + c\right)}{8 b^{3} - c^{3}} \right)}}{12} - \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} - \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right) \log{\left(\frac{- 64 a^{2} b + 384 a b \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} - \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right) - 8 a c^{2} + 16 b^{2} c - 576 b \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} - \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right)^{2} + 24 c^{2} \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} - \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right)}{8 b^{3} - c^{3}} \right)} + \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} - \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right) \log{\left(1 + \frac{- 64 a^{2} b + 384 a b \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} - \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right) - 8 a c^{2} + 16 b^{2} c - 576 b \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} - \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right)^{2} + 24 c^{2} \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} - \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right)}{8 b^{3} - c^{3}} \right)} - \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} + \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right) \log{\left(\frac{- 64 a^{2} b + 384 a b \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} + \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right) - 8 a c^{2} + 16 b^{2} c - 576 b \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} + \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right)^{2} + 24 c^{2} \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} + \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right)}{8 b^{3} - c^{3}} \right)} + \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} + \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right) \log{\left(1 + \frac{- 64 a^{2} b + 384 a b \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} + \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right) - 8 a c^{2} + 16 b^{2} c - 576 b \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} + \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right)^{2} + 24 c^{2} \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} + \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right)}{8 b^{3} - c^{3}} \right)}$$
=
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/ 2 \ / 2 \ / 2 \ / 2 \ / 2 2 2 2 2 \ / 2 2 2 2 2 \
| / ___ \ / ___ \ / ___ \| | / ___ \ / ___ \ / ___ \| | / ___ \ / ___ \ / ___ \| | / ___ \ / ___ \ / ___ \| |- 64*b*a - 8*a*c - 4*b*(c + 2*b + 4*a) + 2*c *(c + 2*b + 4*a) + 16*c*b + 32*a*b*(c + 2*b + 4*a)| | - 64*b*a - 8*a*c - 4*b*(c + 2*b + 4*a) + 2*c *(c + 2*b + 4*a) + 16*c*b + 32*a*b*(c + 2*b + 4*a)|
| | b c a I*\/ 3 *(-c + 2*b)| 2 2 2 2 | b c a I*\/ 3 *(-c + 2*b)| | b c a I*\/ 3 *(-c + 2*b)|| | | b c a I*\/ 3 *(-c + 2*b)| 2 2 2 2 | b c a I*\/ 3 *(-c + 2*b)| | b c a I*\/ 3 *(-c + 2*b)|| | | b c a I*\/ 3 *(-c + 2*b)| 2 2 2 2 | b c a I*\/ 3 *(-c + 2*b)| | b c a I*\/ 3 *(-c + 2*b)|| | | b c a I*\/ 3 *(-c + 2*b)| 2 2 2 2 | b c a I*\/ 3 *(-c + 2*b)| | b c a I*\/ 3 *(-c + 2*b)|| (c + 2*b + 4*a)*log|---------------------------------------------------------------------------------------------------| (c + 2*b + 4*a)*log|1 + ---------------------------------------------------------------------------------------------------|
/ ___ \ | - 576*b*|- -- - -- + - - ------------------| - 64*b*a - 8*a*c + 16*c*b + 24*c *|- -- - -- + - - ------------------| + 384*a*b*|- -- - -- + - - ------------------|| / ___ \ | - 576*b*|- -- - -- + - + ------------------| - 64*b*a - 8*a*c + 16*c*b + 24*c *|- -- - -- + - + ------------------| + 384*a*b*|- -- - -- + - + ------------------|| / ___ \ |- 576*b*|- -- - -- + - - ------------------| - 64*b*a - 8*a*c + 16*c*b + 24*c *|- -- - -- + - - ------------------| + 384*a*b*|- -- - -- + - - ------------------|| / ___ \ |- 576*b*|- -- - -- + - + ------------------| - 64*b*a - 8*a*c + 16*c*b + 24*c *|- -- - -- + - + ------------------| + 384*a*b*|- -- - -- + - + ------------------|| | 3 3 | | 3 3 |
| b c a I*\/ 3 *(-c + 2*b)| | \ 12 24 3 24 / \ 12 24 3 24 / \ 12 24 3 24 /| | b c a I*\/ 3 *(-c + 2*b)| | \ 12 24 3 24 / \ 12 24 3 24 / \ 12 24 3 24 /| | b c a I*\/ 3 *(-c + 2*b)| | \ 12 24 3 24 / \ 12 24 3 24 / \ 12 24 3 24 /| | b c a I*\/ 3 *(-c + 2*b)| | \ 12 24 3 24 / \ 12 24 3 24 / \ 12 24 3 24 /| \ - c + 8*b / \ - c + 8*b /
|- -- - -- + - - ------------------|*log|1 + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------| + |- -- - -- + - + ------------------|*log|1 + ----------------------------------------------------------------------------------------------------------------------------------------------------------------------| - |- -- - -- + - - ------------------|*log|----------------------------------------------------------------------------------------------------------------------------------------------------------------------| - |- -- - -- + - + ------------------|*log|----------------------------------------------------------------------------------------------------------------------------------------------------------------------| - ------------------------------------------------------------------------------------------------------------------------ + ----------------------------------------------------------------------------------------------------------------------------
\ 12 24 3 24 / | 3 3 | \ 12 24 3 24 / | 3 3 | \ 12 24 3 24 / | 3 3 | \ 12 24 3 24 / | 3 3 | 12 12
\ - c + 8*b / \ - c + 8*b / \ - c + 8*b / \ - c + 8*b /
$$- \frac{\left(4 a + 2 b + c\right) \log{\left(\frac{- 64 a^{2} b + 32 a b \left(4 a + 2 b + c\right) - 8 a c^{2} + 16 b^{2} c - 4 b \left(4 a + 2 b + c\right)^{2} + 2 c^{2} \left(4 a + 2 b + c\right)}{8 b^{3} - c^{3}} \right)}}{12} + \frac{\left(4 a + 2 b + c\right) \log{\left(1 + \frac{- 64 a^{2} b + 32 a b \left(4 a + 2 b + c\right) - 8 a c^{2} + 16 b^{2} c - 4 b \left(4 a + 2 b + c\right)^{2} + 2 c^{2} \left(4 a + 2 b + c\right)}{8 b^{3} - c^{3}} \right)}}{12} - \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} - \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right) \log{\left(\frac{- 64 a^{2} b + 384 a b \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} - \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right) - 8 a c^{2} + 16 b^{2} c - 576 b \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} - \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right)^{2} + 24 c^{2} \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} - \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right)}{8 b^{3} - c^{3}} \right)} + \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} - \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right) \log{\left(1 + \frac{- 64 a^{2} b + 384 a b \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} - \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right) - 8 a c^{2} + 16 b^{2} c - 576 b \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} - \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right)^{2} + 24 c^{2} \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} - \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right)}{8 b^{3} - c^{3}} \right)} - \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} + \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right) \log{\left(\frac{- 64 a^{2} b + 384 a b \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} + \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right) - 8 a c^{2} + 16 b^{2} c - 576 b \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} + \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right)^{2} + 24 c^{2} \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} + \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right)}{8 b^{3} - c^{3}} \right)} + \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} + \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right) \log{\left(1 + \frac{- 64 a^{2} b + 384 a b \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} + \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right) - 8 a c^{2} + 16 b^{2} c - 576 b \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} + \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right)^{2} + 24 c^{2} \left(\frac{a}{3} - \frac{b}{12} - \frac{c}{24} + \frac{\sqrt{3} i \left(2 b - c\right)}{24}\right)}{8 b^{3} - c^{3}} \right)}$$
(-b/12 - c/24 + a/3 - i*sqrt(3)*(-c + 2*b)/24)*log(1 + (-576*b*(-b/12 - c/24 + a/3 - i*sqrt(3)*(-c + 2*b)/24)^2 - 64*b*a^2 - 8*a*c^2 + 16*c*b^2 + 24*c^2*(-b/12 - c/24 + a/3 - i*sqrt(3)*(-c + 2*b)/24) + 384*a*b*(-b/12 - c/24 + a/3 - i*sqrt(3)*(-c + 2*b)/24))/(-c^3 + 8*b^3)) + (-b/12 - c/24 + a/3 + i*sqrt(3)*(-c + 2*b)/24)*log(1 + (-576*b*(-b/12 - c/24 + a/3 + i*sqrt(3)*(-c + 2*b)/24)^2 - 64*b*a^2 - 8*a*c^2 + 16*c*b^2 + 24*c^2*(-b/12 - c/24 + a/3 + i*sqrt(3)*(-c + 2*b)/24) + 384*a*b*(-b/12 - c/24 + a/3 + i*sqrt(3)*(-c + 2*b)/24))/(-c^3 + 8*b^3)) - (-b/12 - c/24 + a/3 - i*sqrt(3)*(-c + 2*b)/24)*log((-576*b*(-b/12 - c/24 + a/3 - i*sqrt(3)*(-c + 2*b)/24)^2 - 64*b*a^2 - 8*a*c^2 + 16*c*b^2 + 24*c^2*(-b/12 - c/24 + a/3 - i*sqrt(3)*(-c + 2*b)/24) + 384*a*b*(-b/12 - c/24 + a/3 - i*sqrt(3)*(-c + 2*b)/24))/(-c^3 + 8*b^3)) - (-b/12 - c/24 + a/3 + i*sqrt(3)*(-c + 2*b)/24)*log((-576*b*(-b/12 - c/24 + a/3 + i*sqrt(3)*(-c + 2*b)/24)^2 - 64*b*a^2 - 8*a*c^2 + 16*c*b^2 + 24*c^2*(-b/12 - c/24 + a/3 + i*sqrt(3)*(-c + 2*b)/24) + 384*a*b*(-b/12 - c/24 + a/3 + i*sqrt(3)*(-c + 2*b)/24))/(-c^3 + 8*b^3)) - (c + 2*b + 4*a)*log((-64*b*a^2 - 8*a*c^2 - 4*b*(c + 2*b + 4*a)^2 + 2*c^2*(c + 2*b + 4*a) + 16*c*b^2 + 32*a*b*(c + 2*b + 4*a))/(-c^3 + 8*b^3))/12 + (c + 2*b + 4*a)*log(1 + (-64*b*a^2 - 8*a*c^2 - 4*b*(c + 2*b + 4*a)^2 + 2*c^2*(c + 2*b + 4*a) + 16*c*b^2 + 32*a*b*(c + 2*b + 4*a))/(-c^3 + 8*b^3))/12
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.