Sr Examen
Integral de (A^2)*(t^2)*(cos(2*Pi*f*t+d))^2 dt
Solución
Ha introducido
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1 / | | 2 2 2 | a *t *cos (2*pi*f*t + d) dt | / 0
$$\int\limits_{0}^{1} a^{2} t^{2} \cos^{2}{\left(d + t 2 \pi f \right)}\, dt$$
Integral((a^2*t^2)*cos(((2*pi)*f)*t + d)^2, (t, 0, 1))
Respuesta (Indefinida)
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// / /d \ 3/d \ 2 2 2 3/d \ 2/d \ 4/d \ 3 3 3 4/d \ 3 3 3 2/d \ 2 2 2 /d \ \ \
|| | 3*tan|- + pi*f*t| 3*tan |- + pi*f*t| 3 3 3 24*pi *f *t *tan |- + pi*f*t| 18*pi*f*t*tan |- + pi*f*t| 3*pi*f*t*tan |- + pi*f*t| 8*pi *f *t *tan |- + pi*f*t| 16*pi *f *t *tan |- + pi*f*t| 24*pi *f *t *tan|- + pi*f*t| | |
|| 2 | \2 / \2 / 3*pi*f*t 8*pi *f *t \2 / \2 / \2 / \2 / \2 / \2 / | |
/ ||a *|- ------------------------------------------------------------------- + ------------------------------------------------------------------- + ------------------------------------------------------------------- + ------------------------------------------------------------------- - ------------------------------------------------------------------- - ------------------------------------------------------------------- + ------------------------------------------------------------------- + ------------------------------------------------------------------- + ------------------------------------------------------------------- + -------------------------------------------------------------------| for f != 0|
| || | 3 3 3 3 4/d \ 3 3 2/d \ 3 3 3 3 4/d \ 3 3 2/d \ 3 3 3 3 4/d \ 3 3 2/d \ 3 3 3 3 4/d \ 3 3 2/d \ 3 3 3 3 4/d \ 3 3 2/d \ 3 3 3 3 4/d \ 3 3 2/d \ 3 3 3 3 4/d \ 3 3 2/d \ 3 3 3 3 4/d \ 3 3 2/d \ 3 3 3 3 4/d \ 3 3 2/d \ 3 3 3 3 4/d \ 3 3 2/d \| |
| 2 2 2 || | 48*pi *f + 48*pi *f *tan |- + pi*f*t| + 96*pi *f *tan |- + pi*f*t| 48*pi *f + 48*pi *f *tan |- + pi*f*t| + 96*pi *f *tan |- + pi*f*t| 48*pi *f + 48*pi *f *tan |- + pi*f*t| + 96*pi *f *tan |- + pi*f*t| 48*pi *f + 48*pi *f *tan |- + pi*f*t| + 96*pi *f *tan |- + pi*f*t| 48*pi *f + 48*pi *f *tan |- + pi*f*t| + 96*pi *f *tan |- + pi*f*t| 48*pi *f + 48*pi *f *tan |- + pi*f*t| + 96*pi *f *tan |- + pi*f*t| 48*pi *f + 48*pi *f *tan |- + pi*f*t| + 96*pi *f *tan |- + pi*f*t| 48*pi *f + 48*pi *f *tan |- + pi*f*t| + 96*pi *f *tan |- + pi*f*t| 48*pi *f + 48*pi *f *tan |- + pi*f*t| + 96*pi *f *tan |- + pi*f*t| 48*pi *f + 48*pi *f *tan |- + pi*f*t| + 96*pi *f *tan |- + pi*f*t|| |
| a *t *cos (2*pi*f*t + d) dt = C + |< \ \2 / \2 / \2 / \2 / \2 / \2 / \2 / \2 / \2 / \2 / \2 / \2 / \2 / \2 / \2 / \2 / \2 / \2 / \2 / \2 // |
| || |
/ || 2 3 2 |
|| a *t *cos (d) |
|| ------------- otherwise |
|| 3 |
\\ /
$$\int a^{2} t^{2} \cos^{2}{\left(d + t 2 \pi f \right)}\, dt = C + \begin{cases} a^{2} \left(\frac{8 \pi^{3} f^{3} t^{3} \tan^{4}{\left(\frac{d}{2} + \pi f t \right)}}{48 \pi^{3} f^{3} \tan^{4}{\left(\frac{d}{2} + \pi f t \right)} + 96 \pi^{3} f^{3} \tan^{2}{\left(\frac{d}{2} + \pi f t \right)} + 48 \pi^{3} f^{3}} + \frac{16 \pi^{3} f^{3} t^{3} \tan^{2}{\left(\frac{d}{2} + \pi f t \right)}}{48 \pi^{3} f^{3} \tan^{4}{\left(\frac{d}{2} + \pi f t \right)} + 96 \pi^{3} f^{3} \tan^{2}{\left(\frac{d}{2} + \pi f t \right)} + 48 \pi^{3} f^{3}} + \frac{8 \pi^{3} f^{3} t^{3}}{48 \pi^{3} f^{3} \tan^{4}{\left(\frac{d}{2} + \pi f t \right)} + 96 \pi^{3} f^{3} \tan^{2}{\left(\frac{d}{2} + \pi f t \right)} + 48 \pi^{3} f^{3}} - \frac{24 \pi^{2} f^{2} t^{2} \tan^{3}{\left(\frac{d}{2} + \pi f t \right)}}{48 \pi^{3} f^{3} \tan^{4}{\left(\frac{d}{2} + \pi f t \right)} + 96 \pi^{3} f^{3} \tan^{2}{\left(\frac{d}{2} + \pi f t \right)} + 48 \pi^{3} f^{3}} + \frac{24 \pi^{2} f^{2} t^{2} \tan{\left(\frac{d}{2} + \pi f t \right)}}{48 \pi^{3} f^{3} \tan^{4}{\left(\frac{d}{2} + \pi f t \right)} + 96 \pi^{3} f^{3} \tan^{2}{\left(\frac{d}{2} + \pi f t \right)} + 48 \pi^{3} f^{3}} + \frac{3 \pi f t \tan^{4}{\left(\frac{d}{2} + \pi f t \right)}}{48 \pi^{3} f^{3} \tan^{4}{\left(\frac{d}{2} + \pi f t \right)} + 96 \pi^{3} f^{3} \tan^{2}{\left(\frac{d}{2} + \pi f t \right)} + 48 \pi^{3} f^{3}} - \frac{18 \pi f t \tan^{2}{\left(\frac{d}{2} + \pi f t \right)}}{48 \pi^{3} f^{3} \tan^{4}{\left(\frac{d}{2} + \pi f t \right)} + 96 \pi^{3} f^{3} \tan^{2}{\left(\frac{d}{2} + \pi f t \right)} + 48 \pi^{3} f^{3}} + \frac{3 \pi f t}{48 \pi^{3} f^{3} \tan^{4}{\left(\frac{d}{2} + \pi f t \right)} + 96 \pi^{3} f^{3} \tan^{2}{\left(\frac{d}{2} + \pi f t \right)} + 48 \pi^{3} f^{3}} + \frac{3 \tan^{3}{\left(\frac{d}{2} + \pi f t \right)}}{48 \pi^{3} f^{3} \tan^{4}{\left(\frac{d}{2} + \pi f t \right)} + 96 \pi^{3} f^{3} \tan^{2}{\left(\frac{d}{2} + \pi f t \right)} + 48 \pi^{3} f^{3}} - \frac{3 \tan{\left(\frac{d}{2} + \pi f t \right)}}{48 \pi^{3} f^{3} \tan^{4}{\left(\frac{d}{2} + \pi f t \right)} + 96 \pi^{3} f^{3} \tan^{2}{\left(\frac{d}{2} + \pi f t \right)} + 48 \pi^{3} f^{3}}\right) & \text{for}\: f \neq 0 \\\frac{a^{2} t^{3} \cos^{2}{\left(d \right)}}{3} & \text{otherwise} \end{cases}$$
Respuesta
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/ / 2 2 2 2 \ 2 | 2 |cos (d + 2*pi*f) sin (d + 2*pi*f) sin (d + 2*pi*f) cos (d + 2*pi*f) cos(d + 2*pi*f)*sin(d + 2*pi*f) cos(d + 2*pi*f)*sin(d + 2*pi*f)| a *cos(d)*sin(d) |a *|---------------- + ---------------- - ---------------- + ---------------- - ------------------------------- + -------------------------------| + ---------------- for And(f > -oo, f < oo, f != 0) | | 6 6 2 2 2 2 3 3 4*pi*f | 3 3 | \ 16*pi *f 16*pi *f 32*pi *f / 32*pi *f < | 2 2 | a *cos (d) | ---------- otherwise | 3 \
$$\begin{cases} a^{2} \left(\frac{\sin^{2}{\left(d + 2 \pi f \right)}}{6} + \frac{\cos^{2}{\left(d + 2 \pi f \right)}}{6} + \frac{\sin{\left(d + 2 \pi f \right)} \cos{\left(d + 2 \pi f \right)}}{4 \pi f} - \frac{\sin^{2}{\left(d + 2 \pi f \right)}}{16 \pi^{2} f^{2}} + \frac{\cos^{2}{\left(d + 2 \pi f \right)}}{16 \pi^{2} f^{2}} - \frac{\sin{\left(d + 2 \pi f \right)} \cos{\left(d + 2 \pi f \right)}}{32 \pi^{3} f^{3}}\right) + \frac{a^{2} \sin{\left(d \right)} \cos{\left(d \right)}}{32 \pi^{3} f^{3}} & \text{for}\: f > -\infty \wedge f < \infty \wedge f \neq 0 \\\frac{a^{2} \cos^{2}{\left(d \right)}}{3} & \text{otherwise} \end{cases}$$
=
=
/ / 2 2 2 2 \ 2 | 2 |cos (d + 2*pi*f) sin (d + 2*pi*f) sin (d + 2*pi*f) cos (d + 2*pi*f) cos(d + 2*pi*f)*sin(d + 2*pi*f) cos(d + 2*pi*f)*sin(d + 2*pi*f)| a *cos(d)*sin(d) |a *|---------------- + ---------------- - ---------------- + ---------------- - ------------------------------- + -------------------------------| + ---------------- for And(f > -oo, f < oo, f != 0) | | 6 6 2 2 2 2 3 3 4*pi*f | 3 3 | \ 16*pi *f 16*pi *f 32*pi *f / 32*pi *f < | 2 2 | a *cos (d) | ---------- otherwise | 3 \
$$\begin{cases} a^{2} \left(\frac{\sin^{2}{\left(d + 2 \pi f \right)}}{6} + \frac{\cos^{2}{\left(d + 2 \pi f \right)}}{6} + \frac{\sin{\left(d + 2 \pi f \right)} \cos{\left(d + 2 \pi f \right)}}{4 \pi f} - \frac{\sin^{2}{\left(d + 2 \pi f \right)}}{16 \pi^{2} f^{2}} + \frac{\cos^{2}{\left(d + 2 \pi f \right)}}{16 \pi^{2} f^{2}} - \frac{\sin{\left(d + 2 \pi f \right)} \cos{\left(d + 2 \pi f \right)}}{32 \pi^{3} f^{3}}\right) + \frac{a^{2} \sin{\left(d \right)} \cos{\left(d \right)}}{32 \pi^{3} f^{3}} & \text{for}\: f > -\infty \wedge f < \infty \wedge f \neq 0 \\\frac{a^{2} \cos^{2}{\left(d \right)}}{3} & \text{otherwise} \end{cases}$$
Piecewise((a^2*(cos(d + 2*pi*f)^2/6 + sin(d + 2*pi*f)^2/6 - sin(d + 2*pi*f)^2/(16*pi^2*f^2) + cos(d + 2*pi*f)^2/(16*pi^2*f^2) - cos(d + 2*pi*f)*sin(d + 2*pi*f)/(32*pi^3*f^3) + cos(d + 2*pi*f)*sin(d + 2*pi*f)/(4*pi*f)) + a^2*cos(d)*sin(d)/(32*pi^3*f^3), (f > -oo)∧(f < oo)∧(Ne(f, 0))), (a^2*cos(d)^2/3, True))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.