Evaluate the Integral. Show All Steps

Question

Evaluate the Integral. Show All Steps. If possible attach a video going through the problem

Answer 1

Since two functions are being multiplied, you should think that integration by parts is a probable method to use to find the antiderivative:

Let u = sin(x/4) meaning du = (1/4)cos(x/4)

Let dv = e^2x meaning v = (1/2)e^2x

So because ∫udv = uv - ∫vdu then:

∫e^2xsin(x/4) dx = sin(x/4)•(1/2)e^2x - ∫(1/2)e^2x(1/4)cos(x/4)

= (1/2)e^2xsin(x/4) - (1/8)∫e^2xcos(x/4)

To solve this integral, perform integration by parts a second time:

For (1/8)∫e^2xcos(x/4):

Let u = cos(x/4) meaning du = -(1/4)sin(x/4)

Let dv = e^2x meaning v = (1/2)e^2x

So (1/8)∫e^2xcos(x/4) = (1/8)[cos(x/4)•(1/2)e^2x - ∫(1/2)e^2x•(-(1/4)sin(x/4))]

= (1/16)e^2xcos(x/4) + (1/64)∫e^2xsin(x/4)

Plugging this into the first integral:

∫e^2xsin(x/4) dx = (1/2)e^2xsin(x/4) - (1/16)e^2xcos(x/4) - (1/64)∫e^2xsin(x/4)

Now add (1/64)∫e^2xsin(x/4) to both sides of the equation to get:

(65/64)∫e^2xsin(x/4) dx = (1/2)e^2xsin(x/4) - (1/16)e^2xcos(x/4)

Now multiply both sides of the equation by (64/65) to get:

∫e^2xsin(x/4) dx = (64/65)(1/2)e^2xsin(x/4) - (64/65)(1/16)e^2xcos(x/4)

= (32/65)e^2xsin(x/4) - (4/65)e^2xcos(x/4)

Answer 2

integral of sin(x/4)e^(2x)

= [4(8sin(x/4) - cos(x/4))e^(2x)]/65 + C

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