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Integration by Substitution

Solve: ∫5x-...

Question

Solve: $\int \frac{5 x - 2}{1 + 2 x + 3 x^{2}} . d x$

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Solution

$\int \frac{5 x - 2}{1 + 2 x + 3 x^{2}} d x l e t, u = 5 x - 2 ∴ \int \frac{5 x - 2}{1 + 2 x + 3 x^{2}} d x = \int \frac{5 u}{3 u^{2} + 22 u + 57} d u = 5 \int \frac{u}{3 {(u + \frac{11}{3})}^{2} + \frac{50}{3}} d u a g a i n, l e t v = u + \frac{11}{3} ∴ 5 \int \frac{u}{3 {(u + \frac{11}{3})}^{2} + \frac{50}{3}} d u = 5 \int \frac{3 v - 11}{9 v^{2} + 50} d v s u b s t i t u t i n g, v = \frac{5 \sqrt{2}}{3} w ∴ \int \frac{3 v - 11}{9 v^{2} + 50} d v = 5. \int \frac{5. \sqrt{2} w - 11}{15 \sqrt{2} (w^{2} + 1)} d w = \frac{5}{15 \sqrt{2}} (\int \frac{5 \sqrt{2}}{w^{2} + 1} d w - \int \frac{11}{w^{2} + 1} d w) = \frac{5}{15 \sqrt{2}} [\frac{5}{\sqrt{2}} l o g (\frac{3}{5 \sqrt{2}} {((5 x - 2 + \frac{11}{3}))}^{2}) - 11 arctan \frac{3}{5 \sqrt{2}} ((5 x - 2 + \frac{11}{3}))] + C = \frac{1}{6} [5 l o g (\frac{{(3 x + 1)}^{2}}{2} + 1) - 11 \sqrt{2} arctan (\frac{3 x + 1}{\sqrt{2}})] + C$

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