laplace transform of f(t) = t^2 sin (at) use integration by parts
L{t^2 sin(at)} = (-1)^2 F"(s)
where F(s) = a/(s^2+a^2)
To do the integration by parts, you will have to use multiple parts
u = t^2
dv = sin(at) e^-st
or
u = t^2 sin(at)
dv = e^-st dt
In the first case, you'll need to evaluate v using integration by parts, and the 2nd case will introduce more terms. But in either case, you need to get rid of the powers of t, then integrate by parts twice to get rid of the sin.
To find the Laplace transform of the function f(t) = t^2 sin(at), we can use the technique of integration by parts. The general formula for integration by parts is:
∫ u dv = uv - ∫ v du
Let's break down the problem step by step:
Step 1: Define u and dv
Let u = t^2 and dv = sin(at) dt
Step 2: Calculate du and v
To find du, we differentiate u with respect to t:
du = d/dt(t^2) = 2t
To find v, we integrate dv with respect to t:
v = ∫ sin(at) dt
Step 3: Evaluate v
To integrate sin(at), we can use the substitution method. Let's assign z = at, then dz = a dt.
∫ sin(at) dt = ∫ sin(z) dz
Now, integrating sin(z) with respect to z gives us:
v = -1/a * cos(z) + C = -1/a * cos(at) + C
Step 4: Apply the integration by parts formula
Using the formula for integration by parts, we have:
∫ u dv = uv - ∫ v du
In this case, it becomes:
∫ t^2 sin(at) dt = -t^2/a *cos(at) + ∫ (2t/a * cos(at)) dt
Step 5: Simplify the integral
Now we can simplify the integrals on the right-hand side:
∫ t^2 sin(at) dt = -t^2/a *cos(at) + ∫ (2t/a * cos(at)) dt
= -t^2/a *cos(at) + (2/a) * ∫ (t * cos(at)) dt
Step 6: Repeat the integration by parts
We can apply integration by parts again to the remaining integral:
Let u = t and dv = cos(at) dt.
Step 7: Calculate du and v
To find du, we differentiate u with respect to t:
du = d/dt(t) = 1
To find v, we integrate dv with respect to t:
v = ∫ cos(at) dt = (1/a) * sin(at)
Step 8: Apply the integration by parts formula again
Using the formula for integration by parts again, we have:
∫ t^2 sin(at) dt = -t^2/a *cos(at) + (2/a) * ∫ (t * cos(at)) dt
= -t^2/a *cos(at) + (2/a) * (t * sin(at) - ∫ sin(at) dt)
Step 9: Simplify the integral
Simplifying the remaining integral:
∫ t^2 sin(at) dt = -t^2/a *cos(at) + (2/a) * (t * sin(at) - ∫ sin(at) dt)
= -t^2/a *cos(at) + (2/a) * (t * sin(at) + (-1/a) * cos(at)) + C
This gives us the Laplace transform of f(t) = t^2 sin(at) as follows:
L{f(t)} = -t^2/a *cos(at) + (2/a) * (t * sin(at) + (-1/a) * cos(at)) + C
And that's how we can use integration by parts to find the Laplace transform of f(t) = t^2 sin(at).