Sr Examen
Integral de x/cos^2(3x) dx
Solución
Ha introducido
[src]
pi -- 9 / | | x | --------- dx | 2 | cos (3*x) | / 0
$$\int\limits_{0}^{\frac{\pi}{9}} \frac{x}{\cos^{2}{\left(3 x \right)}}\, dx$$
Integral(x/cos(3*x)^2, (x, 0, pi/9))
Respuesta (Indefinida)
[src]
/ / 2/3*x\\ / /3*x\\ / /3*x\\ 2/3*x\ / /3*x\\ 2/3*x\ / /3*x\\ 2/3*x\ / 2/3*x\\ /3*x\ | log|1 + tan |---|| log|1 + tan|---|| log|-1 + tan|---|| tan |---|*log|1 + tan|---|| tan |---|*log|-1 + tan|---|| tan |---|*log|1 + tan |---|| 6*x*tan|---| | x \ \ 2 // \ \ 2 // \ \ 2 // \ 2 / \ \ 2 // \ 2 / \ \ 2 // \ 2 / \ \ 2 // \ 2 / | --------- dx = C + ------------------ - ----------------- - ------------------ + --------------------------- + ---------------------------- - ---------------------------- - ---------------- | 2 2/3*x\ 2/3*x\ 2/3*x\ 2/3*x\ 2/3*x\ 2/3*x\ 2/3*x\ | cos (3*x) -9 + 9*tan |---| -9 + 9*tan |---| -9 + 9*tan |---| -9 + 9*tan |---| -9 + 9*tan |---| -9 + 9*tan |---| -9 + 9*tan |---| | \ 2 / \ 2 / \ 2 / \ 2 / \ 2 / \ 2 / \ 2 / /
$$\int \frac{x}{\cos^{2}{\left(3 x \right)}}\, dx = C - \frac{6 x \tan{\left(\frac{3 x}{2} \right)}}{9 \tan^{2}{\left(\frac{3 x}{2} \right)} - 9} + \frac{\log{\left(\tan{\left(\frac{3 x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{3 x}{2} \right)}}{9 \tan^{2}{\left(\frac{3 x}{2} \right)} - 9} - \frac{\log{\left(\tan{\left(\frac{3 x}{2} \right)} - 1 \right)}}{9 \tan^{2}{\left(\frac{3 x}{2} \right)} - 9} + \frac{\log{\left(\tan{\left(\frac{3 x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{3 x}{2} \right)}}{9 \tan^{2}{\left(\frac{3 x}{2} \right)} - 9} - \frac{\log{\left(\tan{\left(\frac{3 x}{2} \right)} + 1 \right)}}{9 \tan^{2}{\left(\frac{3 x}{2} \right)} - 9} - \frac{\log{\left(\tan^{2}{\left(\frac{3 x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{3 x}{2} \right)}}{9 \tan^{2}{\left(\frac{3 x}{2} \right)} - 9} + \frac{\log{\left(\tan^{2}{\left(\frac{3 x}{2} \right)} + 1 \right)}}{9 \tan^{2}{\left(\frac{3 x}{2} \right)} - 9}$$
Gráfica
Respuesta
[src]
/ ___\ / ___\
| \/ 3 | | \/ 3 |
log|1 - -----| log|1 + -----| ___
log(4/3) \ 3 / \ 3 / pi*\/ 3
- -------- + -------------- + -------------- + --------
9 9 9 27
$$\frac{\log{\left(1 - \frac{\sqrt{3}}{3} \right)}}{9} - \frac{\log{\left(\frac{4}{3} \right)}}{9} + \frac{\log{\left(\frac{\sqrt{3}}{3} + 1 \right)}}{9} + \frac{\sqrt{3} \pi}{27}$$
=
=
/ ___\ / ___\
| \/ 3 | | \/ 3 |
log|1 - -----| log|1 + -----| ___
log(4/3) \ 3 / \ 3 / pi*\/ 3
- -------- + -------------- + -------------- + --------
9 9 9 27
$$\frac{\log{\left(1 - \frac{\sqrt{3}}{3} \right)}}{9} - \frac{\log{\left(\frac{4}{3} \right)}}{9} + \frac{\log{\left(\frac{\sqrt{3}}{3} + 1 \right)}}{9} + \frac{\sqrt{3} \pi}{27}$$
-log(4/3)/9 + log(1 - sqrt(3)/3)/9 + log(1 + sqrt(3)/3)/9 + pi*sqrt(3)/27
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.