Maharashtra State BoardHSC Commerce 12th Board Exam
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Evaluate the following. ∫2x+6x2+6x+3 dx - Mathematics and Statistics

Sum

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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