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Question
Integrate the following functions with respect to x:
`sqrt(81 + (2x + 1)^2`
Solution
`int sqrt(81 + (2x + 1)^2) "d"x = int sqrt(9^2 + (2x + 1)^2) "d"x`
Put 2x + 1 = t
2 dx = dt
= `int sqrt(9^2 + "t"^2) * 1/2 "dt"`
= `1/2 int sqrt(9^2 + "t"^2) "dt"`
= `1/2 ["t"/2 sqrt("t"^2 + 9^2) + 9^2/2 log |"t" + sqrt("t" + 9^2)|] + "c"`
= `- 1/4 [(2x + 1) sqrt((2x + 1)^2 + 9^2) + 81 log |(2x + 1) + sqrt((2x + 1)^2 + 9^2)|] + "c"`
`int sqrt(81 + (2x + 1)^2) "d"x = 1/4[(2x + 1) sqrt((2x + 1)^2 + 81) + 81 log |(2x + 1) + sqrt((2x + 1)^2 + 81)|] + "c"`
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