Integral of e^(4x)cos(x-6)?
What's the integral of e^(4x)cos(x-6)?
Do we use integration by parts or what? It seems like it goes on forever.
What's the integral of e^(4x)cos(x-6)?
Do we use integration by parts or what? It seems like it goes on forever.
Comments
By integration by parts, we have...
∫ e^(4x)cos(x - 6) dx
u = e^(4x)
du/dx = 4e^(4x)
dv = cos(x - 6) dx
v = sin(x - 6)
You get...
∫ e^(4x)cos(x - 6) dx = e^(4x)sin(x - 6) - 4 ∫ e^(4x)sin(x - 6) dx
Repeat the same steps...
u = e^(4x)
du = 4e^(4x)
dv = sin(x - 6) dx
v = -cos(x - 6)
You get...
∫ e^(4x)cos(x - 6) dx = e^(4x)sin(x - 6) - 4(-e^(4x)cos(x - 6) + ∫ 4e^(4x)cos(x - 6) dx)
∫ e^(4x)cos(x - 6) dx = e^(4x)sin(x - 6) + 4e^(4x)cos(x - 6) - 16 ∫ e^(4x)cos(x - 6) dx
Bring -16 ∫ e^(4x)cos(x - 6) dx to the left to get...
17 ∫ e^(4x)cos(x - 6) dx = e^(4x)sin(x - 6) + 4e^(4x)cos(x - 6)
Divide both sides by 17 and include the constant to get...
∫ e^(4x)cos(x - 6) dx = (e^(4x)sin(x - 6) + 4e^(4x)cos(x - 6))/17 + c
I hope this helps!
Integrate by parts:
int(e^(4x) * cos(x - 6) * dx) = ?
u = e^(4x)
du = 4 * e^(4x) * dx
dv = cos(x - 6) * dx
v = sin(x - 6)
int(e^(4x) * cos(x - 6) * dx) = uv - int(v * du)
int(e^(4x) * cos(x - 6) * dx) = e^(4x) * sin(x - 6) - int(4 * e^(4x) * sin(x - 6) * dx)
int(e^(4x) * cos(x - 6) * dx) = e^(4x) * sin(x - 6) - 4 * int(e^(4x) * sin(x - 6) * dx)
u = e^(4x)
du = 4 * e^(4x) * dx
dv = sin(x - 6) * dx
v = -cos(x - 6)
int(e^(4x) * cos(x - 6) * dx) = e^(4x) * sin(x - 6) - 4 * (uv - int(v * du))
int(e^(4x) * cos(x - 6) * dx) = e^(4x) * sin(x - 6) - 4 * uv + 4 * int(v * du)
int(e^(4x) * cos(x - 6) * dx) = e^(4x) * sin(x - 6) - 4 * e^(4x) * (-cos(x - 6)) + 4 * int(4 * e^(4x) * (-cos(x - 6)) * dx)
int(e^(4x) * cos(x - 6) * dx) = e^(4x) * sin(x - 6) + 4 * e^(4x) * cos(x - 6) - 16 * int(e^(4x) * cos(x - 6) * dx)
int(e^(4x) * cos(x - 6) * dx) + 16 * int(e^(4x) * cos(x - 6) * dx) = e^(4x) * sin(x - 6) + 4 * e^(4x) * cos(x - 6)
17 * int(e^(4x) * cos(x - 6) * dx) = e^(4x) * (sin(x - 6) + 4 * cos(x - 6))
int(e^(4x) * cos(x - 6) * dx) = (1/17) * e^(4x) * (sin(x - 6) + 4 * cos(x - 6))
Add the constant of integration:
int(e^(4x) * cos(x - 6) * dx) = (1/17) * e^(4x) * (sin(x - 6) + 4 * cos(x - 6)) + C
There you go.
e^(4 x) cos(6-x)