Sr Examen
Integral de exp(-px)*cos(4x) dx
Solución
Ha introducido
[src]
1 / | | -p*x | e *cos(4*x) dx | / 0
$$\int\limits_{0}^{1} e^{- p x} \cos{\left(4 x \right)}\, dx$$
Integral(exp((-p)*x)*cos(4*x), (x, 0, 1))
Respuesta (Indefinida)
[src]
// sin(4*x) x*cos(4*x) \
|| -------- - ---------- for p = 0|
|| 16 4 |
|| |
|| // 4*I*x 4*I*x 4*I*x \ |
|| || x*e *sin(4*x) I*e *sin(4*x) I*x*cos(4*x)*e | |
|| || ----------------- - ----------------- + ------------------- for p = -4*I| |
/ || || 2 8 2 | | // x for p = 0\
| || || | | || |
| -p*x || || -4*I*x -4*I*x -4*I*x | | || -p*x |
| e *cos(4*x) dx = C + 4*|< || cos(4*x)*e x*e *sin(4*x) I*x*cos(4*x)*e | | + |<-e |*cos(4*x)
| ||-|<- ---------------- + ------------------ - -------------------- for p = 4*I | | ||------- otherwise|
/ || || 8 2 2 | | || p |
|| || | | \\ /
|| || 4*cos(4*x) p*sin(4*x) | |
|| || - ----------------- - ----------------- otherwise | |
|| || p*x 2 p*x p*x 2 p*x | |
|| || 16*e + p *e 16*e + p *e | |
|| \\ / |
||--------------------------------------------------------------------------------- otherwise|
\\ p /
$$\int e^{- p x} \cos{\left(4 x \right)}\, dx = C + \left(\begin{cases} x & \text{for}\: p = 0 \\- \frac{e^{- p x}}{p} & \text{otherwise} \end{cases}\right) \cos{\left(4 x \right)} + 4 \left(\begin{cases} - \frac{x \cos{\left(4 x \right)}}{4} + \frac{\sin{\left(4 x \right)}}{16} & \text{for}\: p = 0 \\- \frac{\begin{cases} \frac{x e^{4 i x} \sin{\left(4 x \right)}}{2} + \frac{i x e^{4 i x} \cos{\left(4 x \right)}}{2} - \frac{i e^{4 i x} \sin{\left(4 x \right)}}{8} & \text{for}\: p = - 4 i \\\frac{x e^{- 4 i x} \sin{\left(4 x \right)}}{2} - \frac{i x e^{- 4 i x} \cos{\left(4 x \right)}}{2} - \frac{e^{- 4 i x} \cos{\left(4 x \right)}}{8} & \text{for}\: p = 4 i \\- \frac{p \sin{\left(4 x \right)}}{p^{2} e^{p x} + 16 e^{p x}} - \frac{4 \cos{\left(4 x \right)}}{p^{2} e^{p x} + 16 e^{p x}} & \text{otherwise} \end{cases}}{p} & \text{otherwise} \end{cases}\right)$$
Respuesta
[src]
p 4*sin(4) p*cos(4)
------- + ------------- - -------------
2 p 2 p p 2 p
16 + p 16*e + p *e 16*e + p *e
$$- \frac{p \cos{\left(4 \right)}}{p^{2} e^{p} + 16 e^{p}} + \frac{p}{p^{2} + 16} + \frac{4 \sin{\left(4 \right)}}{p^{2} e^{p} + 16 e^{p}}$$
=
=
p 4*sin(4) p*cos(4)
------- + ------------- - -------------
2 p 2 p p 2 p
16 + p 16*e + p *e 16*e + p *e
$$- \frac{p \cos{\left(4 \right)}}{p^{2} e^{p} + 16 e^{p}} + \frac{p}{p^{2} + 16} + \frac{4 \sin{\left(4 \right)}}{p^{2} e^{p} + 16 e^{p}}$$
p/(16 + p^2) + 4*sin(4)/(16*exp(p) + p^2*exp(p)) - p*cos(4)/(16*exp(p) + p^2*exp(p))
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.