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Question
`int (7 + 4x + 5x^2)/(2x + 3)^(3/2) dx`
Solution
Let `I = int(5x^2 + 4x +7)/(2x + 3)^(3/2) dx`
Put 2x + 3 = t2 ...(i)
Differentiating w.r.t. x, we get
2dx = 2t dt
∴ dx = t dt
From (i), we get
`x = (t^2 - 3)/2`
∴ `I = int (5((t^2 - 3)/2)^2 + 4((t^2 - 3)/2) + 7)/(t^2)^(3/2) * t dt`
`I = int (5((t^4 - 6t^2 + 9)/4) + 2t^2 - 6 + 7)/t^3 * t dt`
`I = int (5t^4 - 30t^2 + 45 + 8t^2 + 4)/(4t^3) * t dt`
`I = int (5t^4 - 22t^2 + 49)/(4t^2) dt`
`I = 5/4 int t^2 dt - 22/4 int dt + 49/4 int t^(-2) dt`
`I = 5/4 * t^3/3 - 22/4 t + 49/4 * t^(-1)/(-1) + c`
`I = 5/12t^3 - 11/2t - 49/(4t) + c`
∴ `I = 5/12(2x + 3)^(3/2) - 11/2 sqrt(2x + 3) - 49/4 * 1/sqrt(2x + 3) + c`
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