Maharashtra State BoardHSC Commerce 12th Board Exam
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Evaluate ∫-25x-4-5x-2dx - Mathematics and Statistics

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Sum

Evaluate `int (-2)/(sqrt("5x" - 4) - sqrt("5x" - 2))`dx

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Solution

Let I = `int (-2)/(sqrt("5x" - 4) - sqrt("5x" - 2))`dx

`= -2 int 1/(sqrt("5x" - 4) - sqrt("5x" - 2)) xx (sqrt("5x" - 4) + sqrt("5x" - 2))/(sqrt("5x" - 4) + sqrt("5x" - 2))`dx

`= - 2 int (sqrt("5x" - 4) + sqrt("5x" - 2))/(("5x" - 4) - ("5x" - 2))` dx

`= -2 int (sqrt("5x" - 4) + sqrt("5x" - 2))/-2` dx

`= int [("5x" - 4)^(1/2) + ("5x" - 2)^(1/2)]`dx

`= int ("5x" - 4)^(1/2) "dx" + int ("5x" - 2)^(1/2)` dx

`= ("5x" - 4)^(3/2)/(3/2) xx 1/5 + ("5x" - 2)^(3/2)/(3/2) xx 1/5` + c

∴ I = `2/15 [("5x" - 4)^(3/2) + ("5x" - 2)^(3/2)]` + c

Concept: Methods of Integration: Integration by Substitution
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