Sr Examen
Integral de (cos^5)4x dx
Solución
Ha introducido
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pi -- 8 / | | 5 | cos (x)*4*x dx | / 0
$$\int\limits_{0}^{\frac{\pi}{8}} x 4 \cos^{5}{\left(x \right)}\, dx$$
Integral((cos(x)^5*4)*x, (x, 0, pi/8))
Respuesta (Indefinida)
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/ | 5 5 4 3 2 2 3 | 5 596*cos (x) 32*x*sin (x) 32*sin (x)*cos(x) 208*cos (x)*sin (x) 4 16*x*cos (x)*sin (x) | cos (x)*4*x dx = C + ----------- + ------------ + ----------------- + ------------------- + 4*x*cos (x)*sin(x) + -------------------- | 225 15 15 45 3 /
$$\int x 4 \cos^{5}{\left(x \right)}\, dx = C + \frac{32 x \sin^{5}{\left(x \right)}}{15} + \frac{16 x \sin^{3}{\left(x \right)} \cos^{2}{\left(x \right)}}{3} + 4 x \sin{\left(x \right)} \cos^{4}{\left(x \right)} + \frac{32 \sin^{4}{\left(x \right)} \cos{\left(x \right)}}{15} + \frac{208 \sin^{2}{\left(x \right)} \cos^{3}{\left(x \right)}}{45} + \frac{596 \cos^{5}{\left(x \right)}}{225}$$
Gráfica
Respuesta
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5/2 5/2 2 ___________ 3/2 ___________ 2 3/2
/ ___\ / ___\ / ___\ / ___ / ___\ / ___\ / ___ / ___\ / ___\ / ___\
|1 \/ 2 | |1 \/ 2 | |1 \/ 2 | / 1 \/ 2 |1 \/ 2 | |1 \/ 2 | / 1 \/ 2 |1 \/ 2 | |1 \/ 2 | |1 \/ 2 |
596*|- + -----| 4*pi*|- - -----| 32*|- - -----| * / - + ----- 208*|- + -----| *|- - -----| pi* / - - ----- *|- + -----| 2*pi*|- - -----| *|- + -----|
596 \2 4 / \2 4 / \2 4 / \/ 2 4 \2 4 / \2 4 / \/ 2 4 \2 4 / \2 4 / \2 4 /
- --- + ------------------ + ------------------- + -------------------------------- + ------------------------------ + -------------------------------- + -------------------------------
225 225 15 15 45 2 3
$$- \frac{596}{225} + \frac{4 \pi \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{5}{2}}}{15} + \frac{32 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{15} + \frac{2 \pi \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{3}{2}} \left(\frac{\sqrt{2}}{4} + \frac{1}{2}\right)}{3} + \frac{\pi \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} \left(\frac{\sqrt{2}}{4} + \frac{1}{2}\right)^{2}}{2} + \frac{208 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right) \left(\frac{\sqrt{2}}{4} + \frac{1}{2}\right)^{\frac{3}{2}}}{45} + \frac{596 \left(\frac{\sqrt{2}}{4} + \frac{1}{2}\right)^{\frac{5}{2}}}{225}$$
=
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5/2 5/2 2 ___________ 3/2 ___________ 2 3/2
/ ___\ / ___\ / ___\ / ___ / ___\ / ___\ / ___ / ___\ / ___\ / ___\
|1 \/ 2 | |1 \/ 2 | |1 \/ 2 | / 1 \/ 2 |1 \/ 2 | |1 \/ 2 | / 1 \/ 2 |1 \/ 2 | |1 \/ 2 | |1 \/ 2 |
596*|- + -----| 4*pi*|- - -----| 32*|- - -----| * / - + ----- 208*|- + -----| *|- - -----| pi* / - - ----- *|- + -----| 2*pi*|- - -----| *|- + -----|
596 \2 4 / \2 4 / \2 4 / \/ 2 4 \2 4 / \2 4 / \/ 2 4 \2 4 / \2 4 / \2 4 /
- --- + ------------------ + ------------------- + -------------------------------- + ------------------------------ + -------------------------------- + -------------------------------
225 225 15 15 45 2 3
$$- \frac{596}{225} + \frac{4 \pi \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{5}{2}}}{15} + \frac{32 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{2} \sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}}}{15} + \frac{2 \pi \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)^{\frac{3}{2}} \left(\frac{\sqrt{2}}{4} + \frac{1}{2}\right)}{3} + \frac{\pi \sqrt{\frac{1}{2} - \frac{\sqrt{2}}{4}} \left(\frac{\sqrt{2}}{4} + \frac{1}{2}\right)^{2}}{2} + \frac{208 \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right) \left(\frac{\sqrt{2}}{4} + \frac{1}{2}\right)^{\frac{3}{2}}}{45} + \frac{596 \left(\frac{\sqrt{2}}{4} + \frac{1}{2}\right)^{\frac{5}{2}}}{225}$$
-596/225 + 596*(1/2 + sqrt(2)/4)^(5/2)/225 + 4*pi*(1/2 - sqrt(2)/4)^(5/2)/15 + 32*(1/2 - sqrt(2)/4)^2*sqrt(1/2 + sqrt(2)/4)/15 + 208*(1/2 + sqrt(2)/4)^(3/2)*(1/2 - sqrt(2)/4)/45 + pi*sqrt(1/2 - sqrt(2)/4)*(1/2 + sqrt(2)/4)^2/2 + 2*pi*(1/2 - sqrt(2)/4)^(3/2)*(1/2 + sqrt(2)/4)/3
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.