First slide

Methods of integration

Question

$Let \int \frac{d x}{x^{2008} + x} = \frac{1}{p} \log (\frac{x^{q}}{1 + x^{r}}) + C where p, q, r \in N and need not be$ $distinct then the sum of the digits in p + q + r is$

Moderate

A

8
B

9
C

7
D

2

Solution

$I = \int \frac{1. d x}{x (x^{2007} + 1)} d x = \int (\frac{1}{x} - \frac{x^{2006}}{1 + x^{2007}}) d x$
$\begin{array}{l} = \log x - \frac{1}{2007} \log (1 + x^{2007}) = \frac{\log x^{2007} - \log (1 + x^{2007})}{2007} \\ = \frac{1}{2007} \log (\frac{x^{2007}}{1 + x^{2007}}) + C \\ p + q + r = 6021 \\ The sun of digits in p + q + r is 6 + 0 + 2 + 1 = 9 \end{array}$

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