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Evaluate `Int Tan^(-1) Sqrtx Dx` - ISC (Commerce) Class 12 - Mathematics

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Question

Evaluate `int tan^(-1) sqrtx dx`

Solution

`I = int tan^(-1) sqrtx dx`

Put `sqrtx = t`

`1/(2sqrtx) dx = dt`

`dx = 2sqrtx dt -> dx = 2t dt`

`I = int 2t tan^(-1) t dt`

`I = 2[t^2/2 tan^(-1) t - 1/2 int t^2/(1+ t^2) dt]`

`I = 2[t^2/2 tan^1 t - 1/2 int [(1+t^2)/(1+t^2) - 1/(1+t^2)]dt]`

`I = [t^2 tan^(-1) t - t + tan^(-1) t] + c`

`I = t^2 tan^(-1) t  - t + tan^(-1) t + c`

`I =(x + 1)tan^-1 sqrtx - sqrtx +c `

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Solution Evaluate `Int Tan^(-1) Sqrtx Dx` Concept: Integration as an Inverse Process of Differentiation.
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