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Evaluate the following integrals: inte...

Evaluate the following integrals:
`inte^(3x)sin4xdx`

Text Solution

Verified by Experts

The correct Answer is:
`(e^(3x))/(25)*(3sin4x-4cos4x)+C`
Integrating by parts, taking `e^(3x)` as 2nd function, we get
`I=(sin4x)*(3^(3x))/(3)-int4cos4x*(e^(3x))/(3)dx=-e^(3x)(sin4x)-(4)/(3)*e^(3x)cos4xdx`
`=(1)/(3)e^(3x)(sin4x)-(4)/(3)*{:[(cos4x)*(e^(3x))/(3)-int(-4sin4x)-(e^(3x))/(3)dx]:}`
`=(1)/(3)e^(3x)(sin4x)-(4)/(9)e^(3x)(cos4x)-(16)/(9)I`.
`hArr(1+(16)/(9))I=(1)/(3)e^(3x)(sin4x)-(4)/(9)e^(3x)(cos4x)`.
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