Question

find the value of integration ​

Answer 1

Answer:

$\boxed{\bold{ i = \frac{5}{6} log(3 {x}^{2} + 2x + 1) - \frac{11}{3 \sqrt{2} } {tan}^{ - 1} ( \frac{3x + 1}{ \sqrt{2} } ) + c}}$

For this type of question ,

$\int \: \frac{px + q}{a {x}^{2} + bx + c} dx$

we Let ,

$\green\star \boxed{ \red{ \bold{ (px + q) = A \: (d.c \: of \: (a {x}^{2} + bx + c)) + B }}}$

=> Here , Find the value of A and B by comparing the coefficients of X and constant term on both sides .

=> By this, question will reduce to sum of two integrals .

Formula used :

$\red \star \boxed{ \bold{ \int \frac{1}{x} dx = log(x) + c}}$

$\green\star \boxed{ \bold{ \int \: \frac{1}{ {x}^{2} + a^{2} } dx = \frac{1}{a} {tan}^{ - 1} \frac{x}{a} + c}}$

Solution refer to the attachment