First slide

Methods of integration

Question

The value $\int \frac{d x}{x^{4} + 5 x^{2} + 4}$ IS

Easy

A

$(1 / 3) \tan^{- 1} x - (1 / 2) \tan^{- 1} (x / 2) + C$
B

$3 \tan^{- 1} (x / 3) + 2 \tan^{- 1} (x / 2) + C$
C

$(1 / 3) \tan^{- 1} x + (1 / 6) \tan^{- 1} (x / 2) + C$
D

$(1 / 3) \tan^{- 1} x - (1 / 6) \tan^{- 1} (x / 2) + C$

Solution

$\begin{array}{l} I = \int \frac{d x}{x^{4} + 5 x^{2} + 4} = \int \frac{d x}{(x^{2} + 1) (x^{2} + 4)} \\ = \int \frac{1}{3} [\frac{1}{x^{2} + 1} - \frac{1}{x^{2} + 4}] d x \end{array}$
Therefore, $I = \frac{1}{3} \tan^{- 1} x - \frac{1}{6} \tan^{- 1} \frac{x}{2} + C .$

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