First slide

Methods of integration

Question

$If I = \int \sqrt{\frac{5 - x}{2 + x}} d x, then I equals$

Moderate

A

$\sqrt{x + 2} \sqrt{5 - x} + 3 \sin^{- 1} \sqrt{\frac{x + 2}{3}} + C$
B

$\sqrt{x + 2} \sqrt{5 - x} + 7 \sin^{- 1} \sqrt{\frac{x + 2}{7}} + C$
C

$\sqrt{x + 2} \sqrt{5 - x} + 5 \sin^{- 1} \sqrt{\frac{x + 2}{5}} + C$
D

$None of these$

Solution

$Putting 2 + x = t^{2}, so that d x = 2 t, we get$
$I = \int \frac{\sqrt{7 - t^{2}}}{t} (2 t) d t = 2 \int \sqrt{7 - t^{2}} d t$
$\begin{array}{l} = t \sqrt{7 - t^{2}} + 7 \sin^{- 1} (\frac{t}{\sqrt{7}}) + C \\ ∴ I = \sqrt{x + 2} \sqrt{5 - x} + 7 \sin^{- 1} (\frac{\sqrt{x + 2}}{\sqrt{7}}) + C \end{array}$

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