Sr Examen
Integral de x*ln(ax+b) dx
Solución
Ha introducido
[src]
1 / | | x*log(a*x + b) dx | / 0
$$\int\limits_{0}^{1} x \log{\left(a x + b \right)}\, dx$$
Integral(x*log(a*x + b), (x, 0, 1))
Respuesta (Indefinida)
[src]
/ // x \ \
| || - for a = 0| |
| 2 || b | |
| b *|< | |
| ||log(b + a*x) | |
| 2 ||------------ otherwise| |
| x \\ a / b*x|
a*|--- + ----------------------------- - ---|
/ 2 |2*a 2 2|
| x *log(a*x + b) \ a a /
| x*log(a*x + b) dx = C + --------------- - ---------------------------------------------
| 2 2
/
$$\int x \log{\left(a x + b \right)}\, dx = C - \frac{a \left(\frac{x^{2}}{2 a} + \frac{b^{2} \left(\begin{cases} \frac{x}{b} & \text{for}\: a = 0 \\\frac{\log{\left(a x + b \right)}}{a} & \text{otherwise} \end{cases}\right)}{a^{2}} - \frac{b x}{a^{2}}\right)}{2} + \frac{x^{2} \log{\left(a x + b \right)}}{2}$$
Respuesta
[src]
/ 2 \ 2
log(a + b) | 1 b b *log(a + b)| b *log(b)
---------- - a*|--- - ---- + -------------| + ---------
2 |4*a 2 3 | 2
\ 2*a 2*a / 2*a
$$- a \left(\frac{1}{4 a} - \frac{b}{2 a^{2}} + \frac{b^{2} \log{\left(a + b \right)}}{2 a^{3}}\right) + \frac{\log{\left(a + b \right)}}{2} + \frac{b^{2} \log{\left(b \right)}}{2 a^{2}}$$
=
=
/ 2 \ 2
log(a + b) | 1 b b *log(a + b)| b *log(b)
---------- - a*|--- - ---- + -------------| + ---------
2 |4*a 2 3 | 2
\ 2*a 2*a / 2*a
$$- a \left(\frac{1}{4 a} - \frac{b}{2 a^{2}} + \frac{b^{2} \log{\left(a + b \right)}}{2 a^{3}}\right) + \frac{\log{\left(a + b \right)}}{2} + \frac{b^{2} \log{\left(b \right)}}{2 a^{2}}$$
log(a + b)/2 - a*(1/(4*a) - b/(2*a^2) + b^2*log(a + b)/(2*a^3)) + b^2*log(b)/(2*a^2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.