First slide

Methods of integration

Question

$\int \frac{x (x^{9} + 1)}{x^{2} + 1} d x = f (x) and f (0) = 0 find the value of f (1)$

Moderate

A

$\frac{119}{315} - \frac{π}{4} + \frac{1}{2} \log 2$
B

$\frac{119}{315} - \frac{π}{4} + \log 2$
C

$\frac{283}{315} - \frac{π}{4} + \log 2$
D

$\frac{263}{315} - \frac{π}{4} + \frac{1}{2} \log 2$

Solution

$\int \frac{x (x^{9} + 1)}{x^{2} + 1} d x = \int ((x^{8} - x^{6} + x^{4} - x^{2} + 1) + \frac{x - 1}{1 + x^{2}}) d x$
$\begin{array}{l} f (x) = \frac{x^{9}}{9} - \frac{x^{7}}{7} + \frac{x^{5}}{5} - \frac{x^{3}}{3} + x + \frac{1}{2} \log (1 + x^{2}) - \tan^{- 1} x + C \\ f (0) = 0 \Rightarrow C = 0 \\ ∴ f (1) = \frac{263}{315} - \frac{π}{4} + \frac{1}{2} \log 2 \end{array}$

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