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Question
Integrate the following functions w.r.t. x : `(7 + 4 + 5x^2)/(2x + 3)^(3/2)`
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Solution
Let I = `int(7 + 4x +5x^2)/(2x + 3)^(3/2).dx`
= `int(5x^2 + 4x + 7)/(2x + 3)^(3/2).dx`
Put 2x + 3 = t
∴ 2dx = dt
∴ dx = `dt/(2)`
Also, x = `(t - 3)/(2)`
∴ I = `int(5((t - 3)/2)^2 + 4((t - 3)/2) + 7)/t^(3/2).dt/(2)`
= `(1)/(2) int(5((t^2 - 6t + 9)/4) + 2(t - 3) + 7)/t^(3/2)dt`
= `(1)/(2)int (5t^2 - 30t + 45 + 8t - 24 + 28)/(4t^(3/2))dt`
= `(1)/(8)int(5t^2 - 22t + 49)/t^(3/2)dt`
= `(1)/(8)int(5t^(1/2) - 22t^(-1/2) + 49t^(-3/2))dt`
= `(5)/(8)intt^(1/2)dt - 22/8 int t^(-1/2)dt + 49/8 int t^(-3/2)dt`
= `(5)/(8).t^(3/2)/((3/2)) - (11)/(4).t^(1/2)/((1/2)) + (49)/(8).t^(-1/2)/((-1/2)) + c`
= `(5)/(12)(2x+ 3)^(3/2) - (11)/(2)sqrt(2x + 3) - (49)/(4).(1)/sqrt(2x + 3) + c`.
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