Sr Examen
Integral de (2+sinx)/x^4 dx
Solución
Ha introducido
[src]
oo / | | 2 + sin(x) | ---------- dx | 4 | x | / pi
$$\int\limits_{\pi}^{\infty} \frac{\sin{\left(x \right)} + 2}{x^{4}}\, dx$$
Integral((2 + sin(x))/x^4, (x, pi, oo))
Respuesta (Indefinida)
[src]
/ | / 2\ | 2 + sin(x) 2 Ci(x) log\x / log(x) sin(x) cos(x) sin(x) | ---------- dx = C - ---- - ----- - ------- + ------ - ------ - ------ + ------ | 4 3 6 12 6 3 2 6*x | x 3*x 3*x 6*x | /
$$\int \frac{\sin{\left(x \right)} + 2}{x^{4}}\, dx = C + \frac{\log{\left(x \right)}}{6} - \frac{\log{\left(x^{2} \right)}}{12} - \frac{\operatorname{Ci}{\left(x \right)}}{6} + \frac{\sin{\left(x \right)}}{6 x} - \frac{\cos{\left(x \right)}}{6 x^{2}} - \frac{\sin{\left(x \right)}}{3 x^{3}} - \frac{2}{3 x^{3}}$$
Respuesta
[src]
| / / / 2\ / 2\ / 2\\ / / 2\ / 2\ / 2\\ / 2\ / 2\ / 2\ / 2\ / 2\ / 2\ / 2\ / 2\ / 2\ / 2\ / 2\ / 2\\ |
| | / / pi*I\ \ | | 3*pi | | 5*pi | | pi || | | pi | | 5*pi | | 3*pi || | 3*pi | | pi | | 15*pi | | 9*pi | | 3*pi | | 5*pi | | pi | | pi | | 3*pi | | pi | | 3*pi | / 2\ | 5*pi || |
| | | | ----| | | 160*|12 + -----| 32*|20 + -----| 80*|1 + ---|| | 80*|1 + ---| 32*|20 + -----| 160*|12 + -----|| 640*|-2 - -----| 640*|2 - ---| / 2\ 640*|6 - ------| 640*|-12 + -----| 320*|12 + -----| 64*|20 + -----| 80*|1 + ---| 80*|3 - ---| 320*|24 - -----| / 2\ 320*|1 + ---|*|12 + -----| 64*\-3 + 3*pi /*|20 + -----|| |
| 9/2 | 1280 80 | | 2 | pi*I | |400 \ 2 / \ 2 / \ 2 /| | 400 \ 2 / \ 2 / \ 2 /| \ 4 / \ 4 / 640*\4 + pi / \ 4 / \ 2 / \ 2 / \ 2 / \ 2 / \ 4 / \ 4 / 640*\-4 + pi / \ 2 / \ 2 / \ 2 /| / 2\|
| pi *|- ---- - --- + |- log\pi*e / + ---- + Ci(pi)|*|--- - ---------------- - --------------- + ------------| + |- --- - ------------ + --------------- + ----------------|*EulerGamma - ---------------- + ------------- - ------------- - ---------------- - ----------------- - ---------------- + --------------- + ------------ + ------------ + ---------------- + -------------- - -------------------------- - ----------------------------| 5/2 |pi ||
| 5/2 | 6 4 \ 2 / | 4 4 4 4 | | 4 4 4 4 | 6 6 6 6 6 6 6 4 4 6 6 6 6 | 5/2 5/2 pi *log|---||
| ____ 11*pi \ pi pi \pi 9*pi 3*pi 3*pi / \ pi 3*pi 3*pi 9*pi / pi pi 3*pi 3*pi 9*pi 9*pi 3*pi 3*pi 3*pi 9*pi 3*pi 9*pi 9*pi / pi *EulerGamma pi *log(2) \ 4 /|
|\/ pi - -------- - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------- + ------------ + --------------|
2 | 18 120 3 3 6 |
----- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
3 5/2
3*pi 2*pi
$$- \frac{\left|{- \frac{11 \pi^{\frac{5}{2}}}{18} + \sqrt{\pi} + \frac{\pi^{\frac{5}{2}} \log{\left(\frac{\pi^{2}}{4} \right)}}{6} + \frac{\gamma \pi^{\frac{5}{2}}}{3} + \frac{\pi^{\frac{5}{2}} \log{\left(2 \right)}}{3} - \frac{\pi^{\frac{9}{2}} \left(- \frac{64 \left(-3 + 3 \pi^{2}\right) \left(20 + \frac{5 \pi^{2}}{2}\right)}{9 \pi^{6}} - \frac{320 \left(1 + \frac{\pi^{2}}{2}\right) \left(12 + \frac{3 \pi^{2}}{2}\right)}{9 \pi^{6}} - \frac{640 \left(4 + \pi^{2}\right)}{3 \pi^{6}} - \frac{640 \left(-12 + \frac{9 \pi^{2}}{2}\right)}{9 \pi^{6}} - \frac{1280}{\pi^{6}} - \frac{320 \left(12 + \frac{3 \pi^{2}}{2}\right)}{9 \pi^{6}} - \frac{80}{\pi^{4}} + \frac{640 \left(2 - \frac{\pi^{2}}{4}\right)}{\pi^{6}} + \frac{80 \left(3 - \frac{\pi^{2}}{4}\right)}{3 \pi^{4}} + \frac{320 \left(24 - \frac{3 \pi^{2}}{4}\right)}{9 \pi^{6}} + \frac{64 \left(20 + \frac{5 \pi^{2}}{2}\right)}{3 \pi^{6}} + \frac{640 \left(-4 + \pi^{2}\right)}{3 \pi^{6}} + \frac{80 \left(1 + \frac{\pi^{2}}{2}\right)}{3 \pi^{4}} + \gamma \left(- \frac{400}{\pi^{4}} - \frac{80 \left(1 + \frac{\pi^{2}}{2}\right)}{3 \pi^{4}} + \frac{32 \left(20 + \frac{5 \pi^{2}}{2}\right)}{3 \pi^{4}} + \frac{160 \left(12 + \frac{3 \pi^{2}}{2}\right)}{9 \pi^{4}}\right) - \frac{640 \left(- \frac{3 \pi^{2}}{4} - 2\right)}{\pi^{6}} - \frac{640 \left(6 - \frac{15 \pi^{2}}{4}\right)}{3 \pi^{6}} + \left(\operatorname{Ci}{\left(\pi \right)} - \log{\left(\pi e^{\frac{i \pi}{2}} \right)} + \frac{i \pi}{2}\right) \left(- \frac{160 \left(12 + \frac{3 \pi^{2}}{2}\right)}{9 \pi^{4}} - \frac{32 \left(20 + \frac{5 \pi^{2}}{2}\right)}{3 \pi^{4}} + \frac{80 \left(1 + \frac{\pi^{2}}{2}\right)}{3 \pi^{4}} + \frac{400}{\pi^{4}}\right)\right)}{120}}\right|}{2 \pi^{\frac{5}{2}}} + \frac{2}{3 \pi^{3}}$$
=
=
| / / / 2\ / 2\ / 2\\ / / 2\ / 2\ / 2\\ / 2\ / 2\ / 2\ / 2\ / 2\ / 2\ / 2\ / 2\ / 2\ / 2\ / 2\ / 2\\ |
| | / / pi*I\ \ | | 3*pi | | 5*pi | | pi || | | pi | | 5*pi | | 3*pi || | 3*pi | | pi | | 15*pi | | 9*pi | | 3*pi | | 5*pi | | pi | | pi | | 3*pi | | pi | | 3*pi | / 2\ | 5*pi || |
| | | | ----| | | 160*|12 + -----| 32*|20 + -----| 80*|1 + ---|| | 80*|1 + ---| 32*|20 + -----| 160*|12 + -----|| 640*|-2 - -----| 640*|2 - ---| / 2\ 640*|6 - ------| 640*|-12 + -----| 320*|12 + -----| 64*|20 + -----| 80*|1 + ---| 80*|3 - ---| 320*|24 - -----| / 2\ 320*|1 + ---|*|12 + -----| 64*\-3 + 3*pi /*|20 + -----|| |
| 9/2 | 1280 80 | | 2 | pi*I | |400 \ 2 / \ 2 / \ 2 /| | 400 \ 2 / \ 2 / \ 2 /| \ 4 / \ 4 / 640*\4 + pi / \ 4 / \ 2 / \ 2 / \ 2 / \ 2 / \ 4 / \ 4 / 640*\-4 + pi / \ 2 / \ 2 / \ 2 /| / 2\|
| pi *|- ---- - --- + |- log\pi*e / + ---- + Ci(pi)|*|--- - ---------------- - --------------- + ------------| + |- --- - ------------ + --------------- + ----------------|*EulerGamma - ---------------- + ------------- - ------------- - ---------------- - ----------------- - ---------------- + --------------- + ------------ + ------------ + ---------------- + -------------- - -------------------------- - ----------------------------| 5/2 |pi ||
| 5/2 | 6 4 \ 2 / | 4 4 4 4 | | 4 4 4 4 | 6 6 6 6 6 6 6 4 4 6 6 6 6 | 5/2 5/2 pi *log|---||
| ____ 11*pi \ pi pi \pi 9*pi 3*pi 3*pi / \ pi 3*pi 3*pi 9*pi / pi pi 3*pi 3*pi 9*pi 9*pi 3*pi 3*pi 3*pi 9*pi 3*pi 9*pi 9*pi / pi *EulerGamma pi *log(2) \ 4 /|
|\/ pi - -------- - -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------- + ------------ + --------------|
2 | 18 120 3 3 6 |
----- - ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
3 5/2
3*pi 2*pi
$$- \frac{\left|{- \frac{11 \pi^{\frac{5}{2}}}{18} + \sqrt{\pi} + \frac{\pi^{\frac{5}{2}} \log{\left(\frac{\pi^{2}}{4} \right)}}{6} + \frac{\gamma \pi^{\frac{5}{2}}}{3} + \frac{\pi^{\frac{5}{2}} \log{\left(2 \right)}}{3} - \frac{\pi^{\frac{9}{2}} \left(- \frac{64 \left(-3 + 3 \pi^{2}\right) \left(20 + \frac{5 \pi^{2}}{2}\right)}{9 \pi^{6}} - \frac{320 \left(1 + \frac{\pi^{2}}{2}\right) \left(12 + \frac{3 \pi^{2}}{2}\right)}{9 \pi^{6}} - \frac{640 \left(4 + \pi^{2}\right)}{3 \pi^{6}} - \frac{640 \left(-12 + \frac{9 \pi^{2}}{2}\right)}{9 \pi^{6}} - \frac{1280}{\pi^{6}} - \frac{320 \left(12 + \frac{3 \pi^{2}}{2}\right)}{9 \pi^{6}} - \frac{80}{\pi^{4}} + \frac{640 \left(2 - \frac{\pi^{2}}{4}\right)}{\pi^{6}} + \frac{80 \left(3 - \frac{\pi^{2}}{4}\right)}{3 \pi^{4}} + \frac{320 \left(24 - \frac{3 \pi^{2}}{4}\right)}{9 \pi^{6}} + \frac{64 \left(20 + \frac{5 \pi^{2}}{2}\right)}{3 \pi^{6}} + \frac{640 \left(-4 + \pi^{2}\right)}{3 \pi^{6}} + \frac{80 \left(1 + \frac{\pi^{2}}{2}\right)}{3 \pi^{4}} + \gamma \left(- \frac{400}{\pi^{4}} - \frac{80 \left(1 + \frac{\pi^{2}}{2}\right)}{3 \pi^{4}} + \frac{32 \left(20 + \frac{5 \pi^{2}}{2}\right)}{3 \pi^{4}} + \frac{160 \left(12 + \frac{3 \pi^{2}}{2}\right)}{9 \pi^{4}}\right) - \frac{640 \left(- \frac{3 \pi^{2}}{4} - 2\right)}{\pi^{6}} - \frac{640 \left(6 - \frac{15 \pi^{2}}{4}\right)}{3 \pi^{6}} + \left(\operatorname{Ci}{\left(\pi \right)} - \log{\left(\pi e^{\frac{i \pi}{2}} \right)} + \frac{i \pi}{2}\right) \left(- \frac{160 \left(12 + \frac{3 \pi^{2}}{2}\right)}{9 \pi^{4}} - \frac{32 \left(20 + \frac{5 \pi^{2}}{2}\right)}{3 \pi^{4}} + \frac{80 \left(1 + \frac{\pi^{2}}{2}\right)}{3 \pi^{4}} + \frac{400}{\pi^{4}}\right)\right)}{120}}\right|}{2 \pi^{\frac{5}{2}}} + \frac{2}{3 \pi^{3}}$$
2/(3*pi^3) - Abs(sqrt(pi) - 11*pi^(5/2)/18 - pi^(9/2)*(-1280/pi^6 - 80/pi^4 + (-log(pi*exp_polar(pi*i/2)) + pi*i/2 + Ci(pi))*(400/pi^4 - 160*(12 + 3*pi^2/2)/(9*pi^4) - 32*(20 + 5*pi^2/2)/(3*pi^4) + 80*(1 + pi^2/2)/(3*pi^4)) + (-400/pi^4 - 80*(1 + pi^2/2)/(3*pi^4) + 32*(20 + 5*pi^2/2)/(3*pi^4) + 160*(12 + 3*pi^2/2)/(9*pi^4))*EulerGamma - 640*(-2 - 3*pi^2/4)/pi^6 + 640*(2 - pi^2/4)/pi^6 - 640*(4 + pi^2)/(3*pi^6) - 640*(6 - 15*pi^2/4)/(3*pi^6) - 640*(-12 + 9*pi^2/2)/(9*pi^6) - 320*(12 + 3*pi^2/2)/(9*pi^6) + 64*(20 + 5*pi^2/2)/(3*pi^6) + 80*(1 + pi^2/2)/(3*pi^4) + 80*(3 - pi^2/4)/(3*pi^4) + 320*(24 - 3*pi^2/4)/(9*pi^6) + 640*(-4 + pi^2)/(3*pi^6) - 320*(1 + pi^2/2)*(12 + 3*pi^2/2)/(9*pi^6) - 64*(-3 + 3*pi^2)*(20 + 5*pi^2/2)/(9*pi^6))/120 + pi^(5/2)*EulerGamma/3 + pi^(5/2)*log(2)/3 + pi^(5/2)*log(pi^2/4)/6)/(2*pi^(5/2))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.