If $\int \frac{x^{19} d x}{x^{5} (x^{5} - \sqrt{x^{10} - x^{- 10}})} = \frac{x^{30}}{m} + \frac{(x^{20} - 1)^{3 / 2}}{n} + C,$ where $C$ is arbitrary constant of integration and $m, n \in N,$ then the value of $(m + n)$ is

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Solution

$I = \int \frac{x^{19} d x}{x^{5} (x^{5} - \sqrt{x^{10} - x^{- 10}})}$
$= \int \frac{x^{19} d x}{x^{10} - \sqrt{x^{20} - 1}}$
$= \int x^{19} (x^{10} + \sqrt{x^{20} - 1}) d x$
$= \int (x^{29} + x^{19} \sqrt{x^{20} - 1}) d x$
$= \frac{x^{30}}{30} + \int (x^{19} \sqrt{x^{20} - 1}) d x$
Put $x^{20} - 1 = t^{2} \Rightarrow 20 x^{19} d x = 2 t d t$
$∴ I = \frac{x^{30}}{30} + \frac{1}{10} \int t^{2} d t$
$= \frac{x^{30}}{30} + \frac{1}{30} t^{3} + C$
$= \frac{x^{30}}{30} + \frac{{(x^{20} - 1)}^{3 / 2}}{30} + C$
$∴ m + n = 30 + 30 = 60$

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