INTEGRATION BY PARTS...
1) e^5t sin 3t dt
2) xe^3x dx
3) x sec^2 x dx
**anybody can help me for this question?
if we let A be our integral, we have
A = ∫e^5t sin3t dt
u=e^5t
du = 5e^5t dt
dv = sin3t dt
v = -1/3 cos3t
A = uv - ∫v du
= -1/3 e^5t cos3t + 5/3 ∫e^5t cos3t dt
u=e^5t
du = 5e^5t dt
dv = cos3t dt
v = 1/3 sin3t
A = -1/3 e^5t cos3t + 5/3(1/3 e^5t sin3t - 5/3 ∫e^5t sin3t dt)
= -1/3 e^5t cos3t + 5/9 e^5t sin3t - 25/9 A
34/9 A = 1/9 e^5t (-3cos3t + 5sin3t)
A = 1/34 e^5t (-3cos3t + 5sin3t)
-----------------------------------
A = ∫xe^3x dx
u = x
du = dx
dv = e^3x dx
v = 1/3 e^3x
A = uv - ∫v du
= 1/3 xe^3x - 1/3 ∫e^3x dx
= 1/3 xe^3x - 1/9 e^3x
= 1/9 e^3x (3x-1)
------------------------
A = ∫x sec^2 x dx
u = x
du = dx
dv = sec^2 x dx
v = tan x
A = uv - ∫v du
= x tanx - ∫tanx dx
= x tanx - ∫sinx/cosx dx
= x tanx + log cosx
Sure, I can help you with these integration by parts problems. Integration by parts is a technique used to simplify the integration of products of functions. The formula is given by ∫v du = uv - ∫u dv, where u and v are functions of x.
Let's solve each of the problems one by one:
1) ∫e^(5t) sin(3t) dt:
To solve this, we need to choose u and dv in the integration by parts formula. We generally choose u as the most complicated function and dv as the derivative of the other part.
Let's choose u = sin(3t) and dv = e^(5t) dt.
Now, we will find du and v by taking the derivative and integrating respectively:
du = 3 cos(3t) dt
v = ∫e^(5t) dt = (1/5) e^(5t)
Applying the integration by parts formula, we have:
∫e^(5t) sin(3t) dt = - (1/5)e^(5t) cos(3t) + (3/5)∫e^(5t) cos(3t) dt
Now, we need to solve the integral ∫e^(5t) cos(3t) dt. We can again apply integration by parts to this integral. Repeat the process of choosing u, dv, du, and v as before.
After integrating ∫e^(5t) cos(3t) dt, you will get another integral involving sin(3t). Carry on this process until you can simplify the expression.
2) ∫xe^(3x) dx:
Let's choose u = x and dv = e^(3x) dx.
Now, we will find du and v by taking the derivative and integrating, respectively:
du = dx
v = ∫e^(3x) dx = (1/3) e^(3x)
Using the integration by parts formula, we have:
∫xe^(3x) dx = x * (1/3) e^(3x) - ∫(1/3) e^(3x) dx
You can now simplify the remaining integral, and if necessary, apply the integration by parts formula again until you obtain a simpler expression to integrate.
3) ∫x sec^2(x) dx:
Let's choose u = x and dv = sec^2(x) dx.
Now, we will find du and v by taking the derivative and integrating, respectively:
du = dx
v = ∫sec^2(x) dx = tan(x)
Using the integration by parts formula, we have:
∫x sec^2(x) dx = x * tan(x) - ∫tan(x) dx
The integral of tan(x) can be found by taking the natural logarithm of the absolute value of the secant function.
Hopefully, this step-by-step explanation will help you solve these integration by parts problems.