First slide

Introduction to integration

Question

$If \int \frac{1}{1 + x^{2}} - \frac{1}{(1 + x) 2 \sqrt{x}} d x = - -$

Easy

A

$\cot^{- 1} (\frac{x + \sqrt{x}}{1 - x \sqrt{x}}) + C$
B

$\sin^{- 1} (\sqrt{x}) + C$
C

$\cos^{- 1} (\frac{\sqrt{x} - x}{1 + x \sqrt{x}})$
D

$\tan^{- 1} (\frac{x - \sqrt{x}}{1 + x^{\frac{3}{2}}}) + C$

Solution

$\begin{array}{l} I = \int \frac{1}{1 + x^{2}} - \frac{1}{(1 + x) 2 \sqrt{x}} d x \\ = \int \frac{1}{1 + x^{2}} d x - \int \frac{1}{(1 + (\sqrt{x})^{2})} \frac{1}{2 \sqrt{x}} d x \\ = \int \frac{1}{1 + x^{2}} d x - \int \frac{1}{(1 + t^{2})} d t (where \sqrt{x} = t) \\ = \tan^{- 1} x - \tan^{- 1} \sqrt{x} + C \\ = \tan^{- 1} (\frac{x - \sqrt{x}}{1 + x^{\frac{3}{2}}}) + C \end{array}$

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