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Evaluate: int([sqrt(1+x^2)+x]^n)/(sqrt(1...

Evaluate: `int([sqrt(1+x^2)+x]^n)/(sqrt(1+x^2))dx`

Text Solution

Verified by Experts

The correct Answer is:
`(1)/(n)(sqrt(1+x^(2))+x)^(n)+C`
` "Let " (sqrt(1+x^(2))+x)^(n)=z`
` or n(sqrt(1+x^(2))+x)^(n-1)((x)/(sqrt(1+x^(2)))+1)dx=dz`
` or ((sqrt(1+x^(2))+x)^(n))/(sqrt(1+x^(2)))dx=(dz)/(n)`
` :. " Given integral "=int(dz)/(n)=(1)/(n)z+C=(1)/(n)(sqrt(1+x^(2))+x)^(n)+C`
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