Sr Examen
Integral de 1/(3*sinx+2*cos(x)+3) dx
Solución
Ha introducido
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1 / | | 1 | ----------------------- dx | 3*sin(x) + 2*cos(x) + 3 | / 0
$$\int\limits_{0}^{1} \frac{1}{\left(3 \sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) + 3}\, dx$$
Integral(1/(3*sin(x) + 2*cos(x) + 3), (x, 0, 1))
Respuesta (Indefinida)
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/ / /x\\ / /x\\ | log|1 + tan|-|| log|5 + tan|-|| | 1 \ \2// \ \2// | ----------------------- dx = C + --------------- - --------------- | 3*sin(x) + 2*cos(x) + 3 2 2 | /
$$\int \frac{1}{\left(3 \sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) + 3}\, dx = C + \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{2} - \frac{\log{\left(\tan{\left(\frac{x}{2} \right)} + 5 \right)}}{2}$$
Gráfica
Respuesta
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log(5) log(1 + tan(1/2)) log(5 + tan(1/2)) ------ + ----------------- - ----------------- 2 2 2
$$- \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 5 \right)}}{2} + \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{2} + \frac{\log{\left(5 \right)}}{2}$$
=
=
log(5) log(1 + tan(1/2)) log(5 + tan(1/2)) ------ + ----------------- - ----------------- 2 2 2
$$- \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 5 \right)}}{2} + \frac{\log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{2} + \frac{\log{\left(5 \right)}}{2}$$
log(5)/2 + log(1 + tan(1/2))/2 - log(5 + tan(1/2))/2
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.