Question

Evaluate the following integrals
(a) ${\int }_1^3\frac{1}{(x^2+6x+5)}dx$
(b) ${\int }_1^3(x^2+6x+5)dx$

Answer 1

$\left(a\right){\int }_1^3\frac{1}{\left[\right(x+3{)}^2-2^2]}dx$

$\Rightarrow \frac{1}{2\left(2\right)}\ln {\left(\frac{x+3-2}{x+3+2}\right)}_1^3$

Formula : $\int \frac{1}{x^2-a^2}dx=\frac{1}{2a}ln\left(\frac{x-a}{x+a}\right)$

$\Rightarrow \frac{1}{4}\ln \left(\frac{\frac{4}{8}}{\frac{2}{6}}\right)$

$\Rightarrow \frac{1}{4}\ln \left(\frac{3}{2}\right)$

$\left(b\right){(\frac{x^3}{3}+6\frac{x^2}{2}+5x)}_1^3$

$\Rightarrow (\frac{26}{3}+3(8)+5(2\left)\right)$

$=\left(\frac{70}{3}\right)$