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Question
Solve: `int sqrt(4x^2 + 5)dx`
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Solution
`int sqrt(4x^2 + 5).dx = int sqrt((x^2 + 5/4)).dx`
= `2int sqrt(x^2 + 5/4).dx`
= `2int sqrt(x^2 + (sqrt(5)/2)^2).dx`
= `2[x/2 sqrt(x^2 + 5/4) + (5/4)/2 log|x + sqrt(x^2 + 5/4)|] + c_1`
∵ `int sqrt(x^2 + a^2).dx = x/2 sqrt(x^2 + a^2) + a^2/2 log|x + sqrt(x^2 + a^2)| + c`
= `xsqrt(x^2 + 5/4) + 5/4 log|x + sqrt(x^2 + 5/4)| + c_1`
= `x/2 sqrt(4x^2 + 5) + 5/4 log|x + sqrt((4x^2 + 5)/2)| + c_1`
= `x/2 sqrt(4x^2 + 5) + 5/4 log|(2x + sqrt(4x^2 + 5))/2| + c_1`
= `x/2 sqrt(4x^2 + 5) + 5/4 log|2x + sqrt(4x^2 + 5)| - 5/4 log 2 + c`
= `x/2 sqrt(4x^2 + 5) + 5/4 log|2x + sqrt(4x^2 + 5)| + c_1`
Where c = c1 – `5/4` log2, a constant.
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