First slide

Methods of integration

Question

$If f (x) is a polynomial satisfying f (x) \cdot f (\frac{1}{x}) = f (x) + f (\frac{1}{x}) and f (3) = 82, then \int \frac{f (x)}{x^{2} + 1} d x =$

Moderate

A

$x^{3} - x + 2 \tan^{- 1} x + c$
B

$\frac{1}{3} x^{3} - x + \tan^{- 1} x + c$
C

$\frac{x^{3}}{3} - x + 2 \tan^{- 1} x + c$
D

$\frac{1}{3} x^{3} + x + 2 \tan^{- 1} x + c$

Solution

$\begin{array}{l} Let f (x) = x^{n} + 1 \\ f (3) = 82 \Rightarrow 3^{n} + 1 = 82 \Rightarrow 3^{n} = 3^{4} \\ ∴ n = 4 \\ \int \frac{x^{4} + 1}{x^{2} + 1} d x = \int [x^{2} - 1 + \frac{2}{x^{2} + 1}] d x \end{array}$

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