Evaluate the following. ∫cos5(3z)dzsin2(3z)

Name: Evaluate the following. ∫cos5(3z)dzsin2(3z)
Uploaded: 2022-02-23T17:22:57+00:00
Description: Evaluate the following. ∫cos5(3z)dzsin2(3z)

Question

Evaluate the following.   ∫cos5(3z)dzsin2(3z)

Mitchel Aguirre · Accepted Answer

We have to find the integral of: ∫cos5(3x)sin2(3x)dx  Substitute u=3x⇒dudx=3⇒du3, we obtain ∫cos5(3x)sin2(3x)dx=13∫cos5(3u)sin2(3u)du  Now solving: ∫cos5(3u)sin2(3u)du=∫cos5(U)sin2(U)dU  Preparing for substitution, we use: cos2U=1−sin2U, we obtain  ∫cos5(U)sin2(U)dU  =∫cos⁡(U)(sin2(U)−1)2sin2(U)dU  Substitute v=sin⁡(U)⇒dvdU=cos⁡(u)⇒du=1cos⁡(u)dv, we obtain:  =∫(v2−1)2v2dv  =∫(v2+1v2−2)dv  =∫(v2)dv+∫(1v2)dv−2∫dv  =v33−1v+v  Now, we can undo the substitution v=sin⁡(u), we obtain:  =v33−1v−2v  =sin3(u)3−1sin⁡(u)−2sin⁡(u)  Plug in solved integrals,

Evaluate the following. ∫cos5(3z)dzsin2(3z)

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