Question

\[\int\frac{x + 5}{3 x^2 + 13x - 10}\text{ dx }\]

Answer 1

\[I = \int\frac{x + 5}{3 x^2 + 13x - 10}dx\]
\[ = \int\frac{x + 5}{3 x^2 + 15x - 2x - 10}dx\]
\[ = \int\frac{x + 5}{3x\left( x + 5 \right) - 2\left( x + 5 \right)}dx\]
\[ = \int\frac{x + 5}{\left( 3x - 2 \right)\left( x + 5 \right)}dx\]

\[= \int\frac{x + 5}{(3x - 2)(x + 5)}dx\]
\[ = \int\frac{1}{3x - 2}dx\]
\[ \therefore I = \frac{1}{3}\text{ ln }\left| 3x - 2 \right| + c\]