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Factorization

∫ 2 x+34 x 2+...

Question

$\int (2 x + 3) \sqrt{4 x^{2} + 5 x + 6} d x$

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Solution

$Let I = \int (2 x + 3) \sqrt{4 x^{2} + 5 x + 6} d x & let 2 x + 3 = A \frac{d}{d x} (4 x^{2} + 5 x + 6) + B \Rightarrow 2 x + 3 = A (8 x + 5) + B . . . (1) By equating coefficients of like terms we get, 2 x = 8 A x \Rightarrow A = \frac{1}{4} & 5 A + B = 3 \Rightarrow \frac{5}{4} + B = 3 \Rightarrow B = 3 - \frac{5}{4} = \frac{7}{4} Thus, by substituting the values of A and B in eq (1) we get I = \int (2 x + 3) \sqrt{4 x^{2} + 5 x + 6} d x = \int [\frac{1}{4} (8 x + 5) + \frac{7}{4}] \sqrt{4 x^{2} + 5 x + 6} d x = \frac{1}{4} \int (8 x + 5) \sqrt{4 x^{2} + 5 x + 6} d x + \frac{7}{4} \int \sqrt{4 x^{2} + 5 x + 6} d x Putting 4 x^{2} + 5 x + 6 = t in the first integral \Rightarrow (8 x + 5) d x = d t ∴ I = \frac{1}{4} \int \sqrt{t} \cdot d t + \frac{7 \times 2}{4} \int \sqrt{x^{2} + \frac{5 x}{4} + \frac{3}{2}} d x = \frac{1}{4} \int t^{\frac{1}{2}} \cdot d t + \frac{7}{2} \int \sqrt{x^{2} - \frac{5 x}{4} + {(\frac{5}{8})}^{2} - {(\frac{5}{8})}^{2} + \frac{3}{2}} d x = \frac{1}{4} [\frac{t^{\frac{1}{2} + 1}}{\frac{1}{2} + 1}] + \frac{7}{2} \int \sqrt{{(x + \frac{5}{8})}^{2} - \frac{25}{64} + \frac{3}{2}} d x = \frac{1}{4} \times \frac{2}{3} t^{\frac{3}{2}} + \frac{7}{2} \int \sqrt{{(x + \frac{5}{8})}^{2} + \frac{- 25 + 96}{64}} = \frac{1}{6} t^{\frac{3}{2}} + \frac{7}{2} \int \sqrt{{(x + \frac{5}{8})}^{2} + {(\frac{\sqrt{71}}{8})}^{2}} = \frac{1}{6} {(4 x^{2} + 5 x + 6)}^{\frac{3}{2}} + \frac{7}{2} [\frac{x + \frac{5}{8}}{2} \sqrt{{(x + \frac{5}{8})}^{2} + {(\frac{\sqrt{71}}{8})}^{2}} + \frac{71}{64 \times 2} \ln |x + \frac{5}{8} + \sqrt{{(x + \frac{5}{8})}^{2} + {(\frac{\sqrt{71}}{8})}^{2}}|] + C [∵ \int \sqrt{a^{2} + x^{2}} d x = \frac{1}{2} x \sqrt{a^{2} + x^{2}} + \frac{1}{2} a^{2} \ln |x + \sqrt{x^{2} + a^{2}}| + C] = \frac{1}{6} {(4 x^{2} + 5 x + 6)}^{\frac{3}{2}} + \frac{7}{2} \frac{(8 x + 5)}{16} \sqrt{x^{2} + \frac{5}{4} x + \frac{3}{2}} + \frac{71 \times 7}{2 \times 128} \ln |x + \frac{5}{8} + \sqrt{x^{2} + \frac{5}{4} x + \frac{3}{2}}| + C = \frac{1}{6} {(4 x^{2} + 5 x + 6)}^{\frac{3}{2}} + \frac{7 \times 2 (8 x + 5)}{4 \times 16} \sqrt{x^{2} + \frac{5}{4} x + \frac{3}{2}} + \frac{497}{256} \ln |x + \frac{5}{8} + \sqrt{x^{2} + \frac{5}{4} x + \frac{3}{2}}| + C = \frac{1}{6} (4 x^{2} + 5 x + 6) \sqrt{4 x^{2} + 5 x + 6} + \frac{7}{64} (8 x + 5) \sqrt{4 x^{2} + 5 x + 6} + \frac{497}{256} \ln |x + \frac{5}{6} + \sqrt{x^{2} + \frac{5}{4} x + \frac{3}{2}}| + C = \sqrt{4 x^{2} + 5 x + 6} [\frac{4 x^{2} + 5 x + 6}{6} + \frac{7}{64} (8 x + 5)] + \frac{497}{256} \ln |x + \frac{5}{8} + \sqrt{x^{2} + \frac{5}{4} x + \frac{3}{2}}| + C = \sqrt{4 x^{2} + 5 x + 6} [\frac{128 x^{2} + 328 x + 297}{192}] + \ln |x + \frac{5}{8} + \sqrt{x^{2} + \frac{5}{4} x + \frac{3}{2}}| + C$

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