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Evaluate: inte^(2x)((1+sin2x)/(1+cos2x)...

Evaluate: `inte^(2x)((1+sin2x)/(1+cos2x))dx`

Text Solution

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Given: `inte^(2x)((1+sin2x)/(1+cos2x))dx`
`I=inte^(2x)((1+sin2x)/(1+cos2x))dx`
Let `2x=t implies dx=(dt)/2`
`I=1/2inte^t((1+sint)/(1+cost))dt`
`I=1/2inte^t((1+sint)/(2cos^2(t/2)))dt`
`I=1/2inte^t[1/2sec^2(t/2)+(2sin(t/2)cos(t/2))/(2cos^2(t/2))]dt`
`I=1/2inte^t[1/2sec^2(t/2)+tan(t/2)]dt`
Let `f(t)=tan(t/2)` then `f'(t)=1/2sec^2(t/2)`
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