Sr Examen
Integral de (7+3(sinx/cosx))/(sinx+2cosx)^2 dx
Solución
Ha introducido
[src]
1 / | | sin(x) | 7 + 3*------ | cos(x) | -------------------- dx | 2 | (sin(x) + 2*cos(x)) | / 0
$$\int\limits_{0}^{1} \frac{3 \frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} + 7}{\left(\sin{\left(x \right)} + 2 \cos{\left(x \right)}\right)^{2}}\, dx$$
Integral((7 + 3*(sin(x)/cos(x)))/(sin(x) + 2*cos(x))^2, (x, 0, 1))
Gráfica
Respuesta
[src]
/ / 2 \\ 2 2 / / 2 \\ 2 / / 2 \\
tan(1/2) 6*\pi*I + log\1 - tan (1/2) + tan(1/2)// 6*(pi*I + log(1 - tan(1/2))) 6*log(1 + tan(1/2)) 6*tan (1/2)*(pi*I + log(1 - tan(1/2))) 6*tan (1/2)*log(1 + tan(1/2)) 6*\pi*I + log\1 - tan (1/2) + tan(1/2)//*tan(1/2) 6*tan (1/2)*\pi*I + log\1 - tan (1/2) + tan(1/2)// 6*(pi*I + log(1 - tan(1/2)))*tan(1/2) 6*log(1 + tan(1/2))*tan(1/2)
- ----------------------------- - ---------------------------------------- + ----------------------------- + ----------------------------- - -------------------------------------- - ----------------------------- - ------------------------------------------------- + -------------------------------------------------- + ------------------------------------- + -----------------------------
2 2 2 2 2 2 2 2 2 2
-2 - 2*tan(1/2) + 2*tan (1/2) -2 - 2*tan(1/2) + 2*tan (1/2) -2 - 2*tan(1/2) + 2*tan (1/2) -2 - 2*tan(1/2) + 2*tan (1/2) -2 - 2*tan(1/2) + 2*tan (1/2) -2 - 2*tan(1/2) + 2*tan (1/2) -2 - 2*tan(1/2) + 2*tan (1/2) -2 - 2*tan(1/2) + 2*tan (1/2) -2 - 2*tan(1/2) + 2*tan (1/2) -2 - 2*tan(1/2) + 2*tan (1/2)
$$\frac{6 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{6 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\tan{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{6 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{6 \left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right)}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{6 \left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{6 \left(\log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + \tan{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{6 \left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{6 \left(\log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + \tan{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right) \tan{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{6 \left(\log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + \tan{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right)}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}}$$
=
=
/ / 2 \\ 2 2 / / 2 \\ 2 / / 2 \\
tan(1/2) 6*\pi*I + log\1 - tan (1/2) + tan(1/2)// 6*(pi*I + log(1 - tan(1/2))) 6*log(1 + tan(1/2)) 6*tan (1/2)*(pi*I + log(1 - tan(1/2))) 6*tan (1/2)*log(1 + tan(1/2)) 6*\pi*I + log\1 - tan (1/2) + tan(1/2)//*tan(1/2) 6*tan (1/2)*\pi*I + log\1 - tan (1/2) + tan(1/2)// 6*(pi*I + log(1 - tan(1/2)))*tan(1/2) 6*log(1 + tan(1/2))*tan(1/2)
- ----------------------------- - ---------------------------------------- + ----------------------------- + ----------------------------- - -------------------------------------- - ----------------------------- - ------------------------------------------------- + -------------------------------------------------- + ------------------------------------- + -----------------------------
2 2 2 2 2 2 2 2 2 2
-2 - 2*tan(1/2) + 2*tan (1/2) -2 - 2*tan(1/2) + 2*tan (1/2) -2 - 2*tan(1/2) + 2*tan (1/2) -2 - 2*tan(1/2) + 2*tan (1/2) -2 - 2*tan(1/2) + 2*tan (1/2) -2 - 2*tan(1/2) + 2*tan (1/2) -2 - 2*tan(1/2) + 2*tan (1/2) -2 - 2*tan(1/2) + 2*tan (1/2) -2 - 2*tan(1/2) + 2*tan (1/2) -2 - 2*tan(1/2) + 2*tan (1/2)
$$\frac{6 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{6 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{\tan{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{6 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{6 \left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right)}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{6 \left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{6 \left(\log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + \tan{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{6 \left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{6 \left(\log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + \tan{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right) \tan{\left(\frac{1}{2} \right)}}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{6 \left(\log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + \tan{\left(\frac{1}{2} \right)} + 1 \right)} + i \pi\right)}{-2 - 2 \tan{\left(\frac{1}{2} \right)} + 2 \tan^{2}{\left(\frac{1}{2} \right)}}$$
-tan(1/2)/(-2 - 2*tan(1/2) + 2*tan(1/2)^2) - 6*(pi*i + log(1 - tan(1/2)^2 + tan(1/2)))/(-2 - 2*tan(1/2) + 2*tan(1/2)^2) + 6*(pi*i + log(1 - tan(1/2)))/(-2 - 2*tan(1/2) + 2*tan(1/2)^2) + 6*log(1 + tan(1/2))/(-2 - 2*tan(1/2) + 2*tan(1/2)^2) - 6*tan(1/2)^2*(pi*i + log(1 - tan(1/2)))/(-2 - 2*tan(1/2) + 2*tan(1/2)^2) - 6*tan(1/2)^2*log(1 + tan(1/2))/(-2 - 2*tan(1/2) + 2*tan(1/2)^2) - 6*(pi*i + log(1 - tan(1/2)^2 + tan(1/2)))*tan(1/2)/(-2 - 2*tan(1/2) + 2*tan(1/2)^2) + 6*tan(1/2)^2*(pi*i + log(1 - tan(1/2)^2 + tan(1/2)))/(-2 - 2*tan(1/2) + 2*tan(1/2)^2) + 6*(pi*i + log(1 - tan(1/2)))*tan(1/2)/(-2 - 2*tan(1/2) + 2*tan(1/2)^2) + 6*log(1 + tan(1/2))*tan(1/2)/(-2 - 2*tan(1/2) + 2*tan(1/2)^2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.