Mathematics

Evaluate the following $$\displaystyle \underset{1}{\overset{2}{\int}} \dfrac{5x^2}{x^2 + 4x + 3}$$


SOLUTION
$$\int_{1}^{2}\dfrac{5x^2}{x^2+4x+3}dx$$

$$=5\int_{1}^{2}1+\dfrac{-4x-3}{x^2+4x+3}dx$$

$$=5\int_{1}^{2}1+\dfrac{-4x-3}{(x+2)^2-1}dx$$

$$u=x+2\rightarrow du=dx$$

$$=5\int_{1}^{2}1+\dfrac{-4u+5}{u^2-1}du$$

$$=5\left ( u-2\ln \left | u^2-1 \right |-5\left ( \frac{1}{2}\ln \left | u+1 \right |-\frac{1}{2}\ln \left | u-1 \right | \right ) \right )$$

$$=5\left ( x+2-2\ln \left | (x+2)^2-1 \right |-5\left ( \frac{1}{2}\ln \left | x+2+1 \right |-\frac{1}{2}\ln \left | x+2-1 \right | \right ) \right )$$

$$=5\left ( x+2-2\ln \left | x^2+4x+3\right |-5\left ( \frac{1}{2}\ln \left | x+3 \right |-\frac{1}{2}\ln \left | x+1 \right | \right ) \right )+C$$

$$\int_{1}^{2}\dfrac{5x^2}{x^2+4x+3}dx=5\left ( 4-2\ln (15)-5\frac{\ln 5 -\ln 3}{2}\right )-5\left ( 3+5\dfrac{\ln2}{2}-11\ln 2 \right )$$

$$=5+55\ln 2-10\ln 15+\dfrac{-25\ln\frac{5}{3}-25\ln 2}{2}$$
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Subjective Medium Published on 17th 09, 2020
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