Answer
$$\int\frac{x+4}{x^2+5x-6}dx=\frac{5}{7}\ln|x-1|+\frac{2}{7}\ln|x+6|+C$$
Work Step by Step
$$A=\int\frac{x+4}{x^2+5x-6}dx$$
1) Express the integrand as a sum of partial fractions:
$$\frac{x+4}{x^2+5x-6}=\frac{x+4}{(x-1)(x+6)}=\frac{A}{x-1}+\frac{B}{x+6}$$
Clear fractions:
$$x+4=Ax+6A+Bx-B$$ $$x+4=(A+B)x+(6A-B)$$
Equating coefficients of corresponding powers of $x$, we get
- $A+B=1$
- $6A-B=4$
Calculation gives us $A=5/7$ and $B=2/7$.
Therefore, $$\frac{x+4}{x^2+5x-6}=\frac{5}{7(x-1)}+\frac{2}{7(x+6)}$$
2) Evaluate the integral: $$A=\frac{5}{7}\int\frac{dx}{x-1}+\frac{2}{7}\int\frac{dx}{x+6}$$ $$A=\frac{5}{7}\ln|x-1|+\frac{2}{7}\ln|x+6|+C$$
