Sr Examen
Integral de ln(ax+b)/√(ax+b) dx
Solución
Ha introducido
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1 / | | log(a*x + b) | ------------ dx | _________ | \/ a*x + b | / 0
$$\int\limits_{0}^{1} \frac{\log{\left(a x + b \right)}}{\sqrt{a x + b}}\, dx$$
Integral(log(a*x + b)/sqrt(a*x + b), (x, 0, 1))
Respuesta (Indefinida)
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// _________ _________ \ / ||- 4*\/ a*x + b + 2*\/ a*x + b *log(a*x + b) | | ||-------------------------------------------- for a != 0| | log(a*x + b) || a | | ------------ dx = C + |< | | _________ || x*log(b) | | \/ a*x + b || -------- otherwise | | || ___ | / \\ \/ b /
$$\int \frac{\log{\left(a x + b \right)}}{\sqrt{a x + b}}\, dx = C + \begin{cases} \frac{2 \sqrt{a x + b} \log{\left(a x + b \right)} - 4 \sqrt{a x + b}}{a} & \text{for}\: a \neq 0 \\\frac{x \log{\left(b \right)}}{\sqrt{b}} & \text{otherwise} \end{cases}$$
Respuesta
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/ ___ ___ _______ _______ |4*\/ b - 2*\/ b *log(b) 4*\/ a + b - 2*\/ a + b *log(a + b) |------------------------ - ------------------------------------ for And(a > -oo, a < oo, a != 0) | a a < | log(b) | ------ otherwise | ___ \ \/ b
$$\begin{cases} \frac{- 2 \sqrt{b} \log{\left(b \right)} + 4 \sqrt{b}}{a} - \frac{- 2 \sqrt{a + b} \log{\left(a + b \right)} + 4 \sqrt{a + b}}{a} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\\frac{\log{\left(b \right)}}{\sqrt{b}} & \text{otherwise} \end{cases}$$
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/ ___ ___ _______ _______ |4*\/ b - 2*\/ b *log(b) 4*\/ a + b - 2*\/ a + b *log(a + b) |------------------------ - ------------------------------------ for And(a > -oo, a < oo, a != 0) | a a < | log(b) | ------ otherwise | ___ \ \/ b
$$\begin{cases} \frac{- 2 \sqrt{b} \log{\left(b \right)} + 4 \sqrt{b}}{a} - \frac{- 2 \sqrt{a + b} \log{\left(a + b \right)} + 4 \sqrt{a + b}}{a} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\\frac{\log{\left(b \right)}}{\sqrt{b}} & \text{otherwise} \end{cases}$$
Piecewise(((4*sqrt(b) - 2*sqrt(b)*log(b))/a - (4*sqrt(a + b) - 2*sqrt(a + b)*log(a + b))/a, (a > -oo)∧(a < oo)∧(Ne(a, 0))), (log(b)/sqrt(b), True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.