First slide

Methods of integration

Question

$\int \frac{d x}{\sqrt{2 - 3 x - x^{2}}} = (fog) (x) + C$ then

Moderate

A

$f (x) = \sin^{- 1} x, g (x) = \frac{2 x - 3}{\sqrt{17}}$
B

$\int (x) = \tan^{- 1} x, g (x) = \frac{2 x + 3}{\sqrt{17}}$
C

$f (x) = \sin^{- 1} x, g (x) = \frac{2 x + 3}{\sqrt{17}}$
D

none of these

Solution

$\begin{array}{l} \int \frac{d x}{\sqrt{2 - 3 x - x^{2}}} = \int \frac{d x}{\sqrt{- (x^{2} + 3 x - 2)}} \\ = \int \frac{d x}{\sqrt{- ((x + 3 / 2)^{2} - 17 / 4)}} \\ = \int \frac{d x}{\sqrt{17 / 4 - (x + 3 / 2)^{2}}} \\ = \sin^{- 1} \frac{x + 3 / 2}{\sqrt{17 / 4}} + C = \sin^{- 1} \frac{2 x + 3}{\sqrt{17}} + C \end{array}$
Therefore $g (x) = \frac{2 x + 3}{\sqrt{17}}$ and $f (x) = \sin^{- 1} x$

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