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Question
Evaluate the following.
`int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
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Solution
Let I = `int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
Put x2 + 6x + 3 = t
∴ (2x + 6) dx = dt
∴ I = `int "dt"/sqrt"t"`
`= int "t"^((-1)/2)`dt
`= "t"^(1/2)/(1/2)` + c
`= 2 sqrt"t"` + c
∴ I = `2 sqrt("x"^2 + "6x" + 3)` + c
Alternate Method:
Let I = `int ("2x" + 6)/(sqrt("x"^2 + 6"x" + 3))` dx
`"d"/"dx" ("x"^2 + "6x" + 3)` = 2x + 6
∴ I = `int ("d"/"dx" ("x"^2 + "6x" + 3))/(sqrt("x"^2 + 6"x" + 3))` dx
∴ I = `2 sqrt("x"^2 + "6x" + 3)` + c ....`[because int ("f" '("x"))/sqrt("f"("x")) "dx" = 2sqrt("f"("x")) + "c"]`
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