If-1-x-2-1-x-2-4-dx-A-tan-1-x-B-tan-1-x-2-C-then-

Question Number 86897 by ram roop sharma last updated on 01/Apr/20

If \int \frac{1}{(x^{2} + 1) (x^{2} + 4)} d x

= A \tan^{- 1} x + B \tan^{- 1} \frac{x}{2} + C, then

Commented by Tony Lin last updated on 01/Apr/20

\frac{d}{d x} ({A t a n}^{- 1} x + {B t a n}^{- 1} \frac{x}{2} + C)

= \frac{A}{x^{2} + 1} + \frac{B}{{(\frac{x}{2})}^{2} + 1} \times \frac{1}{2}

= \frac{A}{x^{2} + 1} + \frac{2 B}{x^{2} + 4}

= \frac{A (x^{2} + 4) + 2 B (x^{2} + 1)}{(x^{2} + 1) (x^{2} + 4)}

\Rightarrow {\begin{cases} A + 2 B = 0 \\ 4 A + 2 B = 1 \end{cases}

\Rightarrow A = \frac{1}{3}, B = - \frac{1}{6}

Answered by redmiiuser last updated on 01/Apr/20

A = \frac{1}{3}

B = - \frac{1}{6}

Commented by redmiiuser last updated on 01/Apr/20

s i m p l e w a y t o s o l v e

i s b y p a r t i a l f r a c t i o n s .