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Evaluate int(log(e)(x+sqrt(x^(2)+1)))/(s...

Evaluate `int(log_(e)(x+sqrt(x^(2)+1)))/(sqrt(x^(2)+1))dx.`

Text Solution

Verified by Experts

Since `(d)/(dx)log_(e)(x+sqrt(x^(2)+1))`
`=(1)/(x+sqrt(x^(2)+1))(d)/(dx)(x+sqrt(x^(2)+1))`
`=(1)/(x+sqrt(x^(2)+1))(1+(1)/(sqrt(x^(2)+1)))`
`=(1)/(sqrt(x^(2)+1))`
` :. int(log_(e)(x+sqrt(x^(2)+1)))/(sqrt(x^(2)+1))dx`
`=int log_(e)(x+sqrt(x^(2)+1))(d)/(dx)log_(e)(x+sqrt(x^(2)+1))`
`=((log_(e)(x+sqrt(x^(2)+1)))^(2))/(2)+c`
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