First slide

Methods of integration

Question

$The value of \int \frac{1 + \log x}{\sqrt{{(x^{x})}^{2} - 1}} dx is$

Moderate

A

$\sec^{- 1} (x^{x}) + c$
B

$\tan^{- 1} (x^{x}) + c$
C

$\log (x^{x} + \sqrt{x^{2 x} - 1}) + c$
D

$\cot^{- 1} (x^{x}) + c$

Solution

$\begin{array}{l} I = \int \frac{x^{x} (1 + \log x)}{x^{x} \sqrt{(x^{x}) - 1}} \\ Let x^{x} = t \\ ∴ x \log x = \log t \\ \Rightarrow (\log x + 1) d x = \frac{1}{t} d t \\ \Rightarrow x^{x} (1 + \log x) d x = d t \\ ∴ I = \int \frac{1}{t \sqrt{t^{2} - 1}} d t \\ = \sec^{- 1} (t) = \sec^{- 1} (x^{x}) + C \end{array}$

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