We need to find the integral of (cos x)*(e^2x)

Here integration by parts is the method to be used.

Int [ u dv] = v*u - Int[ v du]

u = cos x, du = -sin x

dv = e^2x , v = e^2x / 2

Int[(cos x)*(e^2x)] = cos...

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We need to find the integral of (cos x)*(e^2x)

Here integration by parts is the method to be used.

Int [ u dv] = v*u - Int[ v du]

u = cos x, du = -sin x

dv = e^2x , v = e^2x / 2

Int[(cos x)*(e^2x)] = cos x*e^2x / 2 + Int[ sin x*e^2x]/2

Again use integration by parts for Int[ sin x*e^2x]/2

u = sin x , du = cos x

dv = e^2x, v = e^2x / 2

Int[ sin x*e^2x]/2 = sin x*e^2x / 4 - Int [cos x e^2x] / 4

So we get Int[(cos x)*(e^2x)] = (4/5)[cos x*e^2x / 2 +sin x*e^2x / 4]

=> (2/5)*cos x*e^2x + (1/5)*sin x*e^2x + C

**The integral of (cos x)*(e^2x) is (2/5)*cos x*e^2x + (1/5)*sin x*e^2x + C**