Let f(x) = ln \textrm{ } \left|\right. \textrm{ } x \textrm{ } + \textrm{ } \sqrt{x^{2} \textrm{ } + \textrm{ } 1} \textrm{ } \left|\right.. Then the value of \int_{- 5}^{5} \frac{dx}{\left(\right. 1 + f^{5} \left(\right. x \left.\right) \left.\right) \textrm{ } + \textrm{ } \sqrt{\left(\right. 1 + f^{10} \left(\right. x \left.\right) \left.\right)}} \textrm{ } dx is

Engineering

Mathematics

Properties of Definite Integral

Question

Let f(x) = $\ln | x + \sqrt{x^{2} + 1} |$ . Then the value of $\int_{- 5}^{5} \frac{dx}{(1 + f^{5} (x)) + \sqrt{(1 + f^{10} (x))}} dx$ is

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f(– x) = – f(x)

$\int_{- 5}^{5} \frac{(1 + f^{5} (x)) - \sqrt{1 + f^{10} (x)}}{{(1 + f^{5} (x))}^{2} - (1 + f^{10} (x))} dx = \int_{- 5}^{5} \frac{1 + f^{5} (x) - \sqrt{1 + f^{10} (x)}}{2 f^{5} (x)} dx$

$= \int_{- 5}^{5} (\frac{1}{2 f^{5}} + \frac{1}{2} - \frac{\sqrt{1 + f^{10} (x)}}{2 f^{5} (x)}) dx$

As f(x) is an odd function.

$= (0 + \frac{10}{2} - 0) = \frac{10}{2} = 5$