First slide

Introduction to integration

Question

If $f (x) = lim_{n \to \infty} \frac{x^{n} - x^{- n}}{x^{n} + x^{- n}}, x > 1$ then
$\int \frac{x f (x) \log (x + \sqrt{1 + x^{2}})}{\sqrt{1 + x^{2}}}$ dx is

Moderate

A

$\log (x + \sqrt{1 + x^{2}}) - x + C$
B

$(\frac{1}{2}) x^{2} (\log (x + \sqrt{1 + x^{2}}) - 1) + C$
C

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

none of these

Solution

$f^{'} (x) = lim_{n \to \infty} \frac{x^{n} - x^{- n}}{x^{n} + x^{- n}} = lim_{n \to \infty} \frac{1 - (1 / x)^{2 n}}{1 + (1 / x)^{2 n}} = 1$
$(1 / x < 1)$
Using integration by parts , we get
$\begin{array}{l} I = \int \frac{x \log (x + \sqrt{1 + x^{2}})}{\sqrt{1 + x^{2}}} d x \\ = \sqrt{1 + x^{2}} \log (x + \sqrt{1 + x^{2}}) d x - \int d x \\ = \sqrt{1 + x^{2}} \log (x + \sqrt{1 + x^{2}}) - x + C . \end{array}$

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