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Evaluate: inte^x\ ((sin4x-4)/(1-cos4x))\...

Evaluate: `inte^x\ ((sin4x-4)/(1-cos4x))\ dx`

Text Solution

Verified by Experts

Let
`I=inte^x\ ((sin4x-4)/(1-cos4x))\ dx`
`=int e^x((2sin(2x)cos(2x)-4)/(2 sin^2(2x)))dx`
`=int e^x((2sin(2x)cos(2x))/(2 sin^2(2x))-4/(2 sin^2(2x))dx`
`=int e^x (cot(2x)-2cosec^2(2x))dx`
where, `f(x)=cot(2x)` and `f'(x)=-2cosec^(2x)`
`=inte^x [f(x)+f'(x)]dx`
`=e^xf(x)+c`
`I=e^xcot(2x)+c`
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