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Question
Evaluate the definite integral:
`int_0^1 (2x + 3)/(5x^2 + 1) dx`
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Solution
Let `I = int_0^1 (2x + 3)/(5x^2 + 1) dx`
`= int_0^1 (2x)/(5x^2 + 1) dx + int_0^1 3/(5x^2 + 1) dx`
`= 1/5 int_0^1 (10 x)/(5x^2 + 1) dx + 3/5 int_0^1 1/(x^2 + 1/sqrt5) dx`
`= 1/5 [log (5x^2 + 1)]_0^1 + 3/5 . 1/(1 sqrt5) [tan^-1 (x/ 1)/sqrt5]_0^1`
`= 1/5 [log 6 - log 1] + 3/sqrt5 [tan^-1 sqrt5 - tan^-1 0]`
`= 1/5 log 6 + 3/sqrt5 tan^-1 sqrt5`
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