Sr Examen
Integral de sin(pinx/2) dx
Solución
Ha introducido
[src]
pi / | | /p*log(x)\ | sin|--------| dx | \ 2 / | / 0
$$\int\limits_{0}^{\pi} \sin{\left(\frac{p \log{\left(x \right)}}{2} \right)}\, dx$$
Integral(sin((p*log(x))/2), (x, 0, pi))
Respuesta (Indefinida)
[src]
// 2 \
|| I*log(x) I*x |
|| -------- - ---- for p = -2*I|
|| 2 4 |
|| |
/ || 2 |
| || I*log(x) I*x |
| /p*log(x)\ || - -------- + ---- for p = 2*I |
| sin|--------| dx = C + |< 2 4 |
| \ 2 / || |
| || /p*log(x)\ 2/p*log(x)\ |
/ || 8*x*tan|--------| 2*p*x*tan |--------| |
|| 2*p*x \ 4 / \ 4 / |
||- --------------------------------------------- + --------------------------------------------- + --------------------------------------------- otherwise |
|| 2 2/p*log(x)\ 2 2/p*log(x)\ 2 2/p*log(x)\ 2 2/p*log(x)\ 2 2/p*log(x)\ 2 2/p*log(x)\ |
|| 4 + p + 4*tan |--------| + p *tan |--------| 4 + p + 4*tan |--------| + p *tan |--------| 4 + p + 4*tan |--------| + p *tan |--------| |
\\ \ 4 / \ 4 / \ 4 / \ 4 / \ 4 / \ 4 / /
$$\int \sin{\left(\frac{p \log{\left(x \right)}}{2} \right)}\, dx = C + \begin{cases} - \frac{i x^{2}}{4} + \frac{i \log{\left(x \right)}}{2} & \text{for}\: p = - 2 i \\\frac{i x^{2}}{4} - \frac{i \log{\left(x \right)}}{2} & \text{for}\: p = 2 i \\\frac{2 p x \tan^{2}{\left(\frac{p \log{\left(x \right)}}{4} \right)}}{p^{2} \tan^{2}{\left(\frac{p \log{\left(x \right)}}{4} \right)} + p^{2} + 4 \tan^{2}{\left(\frac{p \log{\left(x \right)}}{4} \right)} + 4} - \frac{2 p x}{p^{2} \tan^{2}{\left(\frac{p \log{\left(x \right)}}{4} \right)} + p^{2} + 4 \tan^{2}{\left(\frac{p \log{\left(x \right)}}{4} \right)} + 4} + \frac{8 x \tan{\left(\frac{p \log{\left(x \right)}}{4} \right)}}{p^{2} \tan^{2}{\left(\frac{p \log{\left(x \right)}}{4} \right)} + p^{2} + 4 \tan^{2}{\left(\frac{p \log{\left(x \right)}}{4} \right)} + 4} & \text{otherwise} \end{cases}$$
Respuesta
[src]
/p*log(pi)\ /p*log(pi)\
4*pi*sin|---------| 2*pi*p*cos|---------|
\ 2 / \ 2 /
------------------- - ---------------------
2 2
4 + p 4 + p
$$- \frac{2 \pi p \cos{\left(\frac{p \log{\left(\pi \right)}}{2} \right)}}{p^{2} + 4} + \frac{4 \pi \sin{\left(\frac{p \log{\left(\pi \right)}}{2} \right)}}{p^{2} + 4}$$
=
=
/p*log(pi)\ /p*log(pi)\
4*pi*sin|---------| 2*pi*p*cos|---------|
\ 2 / \ 2 /
------------------- - ---------------------
2 2
4 + p 4 + p
$$- \frac{2 \pi p \cos{\left(\frac{p \log{\left(\pi \right)}}{2} \right)}}{p^{2} + 4} + \frac{4 \pi \sin{\left(\frac{p \log{\left(\pi \right)}}{2} \right)}}{p^{2} + 4}$$
4*pi*sin(p*log(pi)/2)/(4 + p^2) - 2*pi*p*cos(p*log(pi)/2)/(4 + p^2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.