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Question
Integrate the function `1/sqrt(1+4x^2)`
Solution
Let `I = int 1/sqrt(1 + 4x^2) dx`
`= 1/2 int dx/sqrt(1/4 + x)`
`= 1/2 int dx/sqrt((1/2)^2 + x^2)`
`= 1/2 log abs (x sqrt(1/4 +x^2)) C_1`
.....`[because int dx/sqrt(x^2 + a^2) = log abs (x + sqrt(x^2 + a^2)) + C]`
`= 1/2 log abs ((2x + sqrt(1 + 4x^2))/2) + C_1`
`= 1/2 log abs (2x + sqrt(1 + 4x^2)) - 1/2 log 2 + C_1`
`= 1/2 log |2x + sqrt 1 + 4x^2| + C` .... [`C = -1/2 log 2 + C_1`]
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