Question

Integrate the function: $\frac{5x}{(x+1)(x^2+9)}$

Answer 1

Let \frac{5x}{(x+1)(x^2+9)}$$=\frac{A}{(x+1)}+\frac{Bx+C}{(x^2+9)}...(1)

⇒5x=A(x2+9)+(Bx+C)(x+1)

⇒5x=Ax2+9A+Bx+Cx+C

Equating the coefficients of x2,x,and constant term,we obtain

A+B=0

B+C=5

9A+C=0

On solving these equations,we obtain

A=$-\frac{1}{2}$,B=$\frac{1}{2}$,and C=$\frac{9}{2}$

From equation(1),we obtain

$\frac{5x}{(x+1)(x^2+9)}$=$-\frac{1}{2}$(x+1)+$\frac{\frac{x}{2}+\frac{9}{2}}{x^2+9}$

=$-\frac{1}{2}$log|x+1|+$\frac{1}{2}$∫$\frac{x}{x^2+9}$dx+$\frac{9}{2}$∫$\frac{1}{x^2+9}$dx

=$-\frac{1}{2}$log|x+1|+$\frac{1}{4}$log(x2+9)+$\frac{3}{2}$tan-1$\frac{x}{3}$+C