Verified answer

let a = - 1 , b = 9

∫e^(-Θ)cos(9Θ)dΘ

...e^(-Θ)[-cos(9Θ) + 9sin(9Θ)]

=-------------------------------------------

................(-1)^2 + 9^2

.....- e^(-Θ)[cos(9Θ) - 9sin(9Θ)]

=---------------------------------------------- + C answer//

..................82

Let X = ∫e^(-θ)cos(9θ) dθ. The key step is to recognize X after integrating by parts twice.

Let u = e^(-θ) and dv = cos(9θ)dθ.

Then du = -e^(-θ)dθ and v = (1/9)sin(9θ).

So by integration by parts, we have:

X

= uv - ∫vdu

= (1/9)e^(-θ)sin(9θ) - ∫-(1/9)e^(-θ)sin(9θ)dθ

= (1/9)e^(-θ)sin(9θ) + ∫(1/9)e^(-θ)sin(9θ)dθ

Let f = (1/9)e^(-θ) and dg = sin(9θ)dθ.

Then df = (-1/9)e^(-θ)dθ and g = (-1/9)cos(9θ).

So by integration by parts, we have:

X

= (1/9)e^(-θ)sin(9θ) + ∫(1/9)e^(-θ)sin(9θ)dθ

= (1/9)e^(-θ)sin(9θ) + fg - ∫gdf

= (1/9)e^(-θ)sin(9θ) - (1/81)e^(-θ)cos(9θ) - ∫(1/81)e^(-θ)cos(9θ)dθ

= (1/9)e^(-θ)sin(9θ) - (1/81)e^(-θ)cos(9θ) - (1/81)X

Solving for X, we obtain:

X = (1/9)e^(-θ)sin(9θ) - (1/81)e^(-θ)cos(9θ) - (1/81)X

(82/81)X = (1/9)e^(-θ)sin(9θ) - (1/81)e^(-θ)cos(9θ)

X = (9/82)e^(-θ)sin(9θ) - (1/82)e^(-θ)cos(9θ) + C

You can always check an integration answer by differentiating it. That is the right answer.

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