Question

Integrate by rational fraction:
$ln(x \div \sqrt{(x + 1)(x + 2)} )$
​

Answer 1

Step-by-step explanation:

$\huge\sf\red{Let}$

$\huge\frac{x}{(x-1)(x-2)}=\frac{A}{x+1}+\frac{B}{x+2}$ eqn.......1

$\red\implies\frac{x}{(x+1)(x+2)}=\frac{A(x+2)+B(x+1)}{(x+1)(x+2)} \\ \\ \implies\:x=A(x+2)+B(x+1)$

when x = -1,

-1 = A(-1+2)+B(-1+1) , A = -1

when x= -2

-2 = A(-2+2)+B(-2+1) , B = 2

From 1

$\sf\int\frac{x}{(x+1)(x+2)}dx=\int\frac{-1}{x+1}dx+\int\frac{2}{x+2}dx$

= - $\bf\int\frac{1}{x+1}dx+2\int\frac{1}{x+2}dx$

= log$\sf\mod{{x+2}^{2}}- log\mod{x+1}+C$

=log$\bf\mod\frac{{x+2}^{2}}{x+1}+C$