Sr Examen
Resolución de integrales/
((b^2-a^2)*cos(x)*sin(x)+(a^3*b+b^3*a)*(cos(x)*sin(x))^2+a*b*(b^2*sin(x)^4+a^2*cos(x)^4))/(a^2*cos(x)^2+b^2*sin(x)^2)
Integral de ((b^2-a^2)*cos(x)*sin(x)+(a^3*b+b^3*a)*(cos(x)*sin(x))^2+a*b*(b^2*sin(x)^4+a^2*cos(x)^4))/(a^2*cos(x)^2+b^2*sin(x)^2) dx
Solución
Ha introducido
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1 / | | / 2 2\ / 3 3 \ 2 / 2 4 2 4 \ | \b - a /*cos(x)*sin(x) + \a *b + b *a/*(cos(x)*sin(x)) + a*b*\b *sin (x) + a *cos (x)/ | ---------------------------------------------------------------------------------------- dx | 2 2 2 2 | a *cos (x) + b *sin (x) | / 0
$$\int\limits_{0}^{1} \frac{a b \left(a^{2} \cos^{4}{\left(x \right)} + b^{2} \sin^{4}{\left(x \right)}\right) + \left(\left(- a^{2} + b^{2}\right) \cos{\left(x \right)} \sin{\left(x \right)} + \left(\sin{\left(x \right)} \cos{\left(x \right)}\right)^{2} \left(a^{3} b + a b^{3}\right)\right)}{a^{2} \cos^{2}{\left(x \right)} + b^{2} \sin^{2}{\left(x \right)}}\, dx$$
Integral((((b^2 - a^2)*cos(x))*sin(x) + (a^3*b + b^3*a)*(cos(x)*sin(x))^2 + (a*b)*(b^2*sin(x)^4 + a^2*cos(x)^4))/(a^2*cos(x)^2 + b^2*sin(x)^2), (x, 0, 1))
Respuesta
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/ / 2 2 \ | | 2 b *sin (1)| |log|cos (1) + ----------| | | 2 | < \ a / 2 2 |------------------------- + a*b*cos (1) + a*b*sin (1) for And(a > -oo, a < oo, a != 0) | 2 | \ oo otherwise
$$\begin{cases} a b \cos^{2}{\left(1 \right)} + a b \sin^{2}{\left(1 \right)} + \frac{\log{\left(\cos^{2}{\left(1 \right)} + \frac{b^{2} \sin^{2}{\left(1 \right)}}{a^{2}} \right)}}{2} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\\infty & \text{otherwise} \end{cases}$$
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/ / 2 2 \ | | 2 b *sin (1)| |log|cos (1) + ----------| | | 2 | < \ a / 2 2 |------------------------- + a*b*cos (1) + a*b*sin (1) for And(a > -oo, a < oo, a != 0) | 2 | \ oo otherwise
$$\begin{cases} a b \cos^{2}{\left(1 \right)} + a b \sin^{2}{\left(1 \right)} + \frac{\log{\left(\cos^{2}{\left(1 \right)} + \frac{b^{2} \sin^{2}{\left(1 \right)}}{a^{2}} \right)}}{2} & \text{for}\: a > -\infty \wedge a < \infty \wedge a \neq 0 \\\infty & \text{otherwise} \end{cases}$$
Piecewise((log(cos(1)^2 + b^2*sin(1)^2/a^2)/2 + a*b*cos(1)^2 + a*b*sin(1)^2, (a > -oo)∧(a < oo)∧(Ne(a, 0))), (oo, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.