Mathematics

Solve:
$$\displaystyle\int {\dfrac{{4x + 6}}{{2{x^2} + 5x + 3}}dx} $$


SOLUTION
$$\displaystyle I= \int \frac {4x+6}{2x^2+5x+3} dx= \int \frac {4x+6 dx}{2x^2+3x+2x+3}= \int \frac {(4x+6)dx}{(2x+3)x+1(2x+3)}$$

$$\displaystyle I = \int \frac {dx(4x+6)}{(x+1)(2x+3)}=\int \frac {dx(4x+4)}{(x+1)(2x+3)}+2 \int \frac {dx}{(x+1)(2x+3)}$$

$$=\displaystyle \int \frac {4dx}{(2x+3)}+ 2\int \frac{dx}{(x+1)(2x+3)}$$

$$\displaystyle I = a |n|2x+3| = 2 \left [ \frac {2}{(2x+3)}- \frac {1}{(x+1)} \right ]$$

$$\displaystyle I = 4|n|2x+3| - 4|n|2x+3|+ 2|n|x+1|+ |n|c|$$

$$\displaystyle I = |n|c(x+1)^2|$$

$$\displaystyle \therefore \int \frac {4x+6}{(2x^2+5x+3)}dx = |n|c(x+1)^2|$$     c= constant.
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Subjective Medium Published on 17th 09, 2020
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