First slide

Methods of integration

Question

If $\int \frac{\log (x + \sqrt{1 + x^{2}})}{\sqrt{1 + x^{2}}} dx = gof (x) +$ constant, then

Moderate

A

$f (x) = \log (x + \sqrt{x^{2} + 1})$
B

$f (x) = \log (x + \sqrt{x^{2} + 1}) and g (x) = x^{2}$
C

$f (x) = \log (x + \sqrt{x^{2} + 1}) and g (x) = \frac{x^{2}}{2}$
D

$f (x) = \frac{x^{2}}{2} and g (x) = \log (x + \sqrt{x^{2} + 1})$

Solution

On putting $\log (x + \sqrt{1 + x^{2}}) = t$
$\Rightarrow \frac{1}{x + \sqrt{1 + x^{2}}} \times (1 + \frac{x}{\sqrt{1 + x^{2}}}) dx = dt$
$\begin{array}{l} \Rightarrow \frac{dx}{\sqrt{1 + x^{2}}} = dt \\ ∴ \int \frac{\log (x + \sqrt{1 + x^{2}})}{\sqrt{1 + x^{2}}} dx = \int tdt = \frac{1}{2} t^{2} + C \\ = \frac{1}{2} {[\log (x + \sqrt{x^{2} + 1})]}^{2} + C \end{array}$

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