Find: ∣∫dx9−4x2

Given:∫dx9−4x2=∫dx494−x2=∫dx294−x2=12∫dx94−x2=12∫dx322−x2We can see that it is the form of ∫dxa2−x2=sin−1 xa+CHence, replacing a by 32, we will get=12sin−1 2x3+C∴∫dx9-4x2=12sin−1 2x3+CWhich is the required answer.

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Find: $∣ \int \frac{dx}{\sqrt{9 - 4 x^{2}}}$

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Detailed Solution

Given:
$\int \frac{dx}{\sqrt{9 - 4 x^{2}}}$
$\begin{array}{l} = \int \frac{dx}{\sqrt{4 (\frac{9}{4} - x^{2})}} \\ = \int \frac{dx}{2 \sqrt{(\frac{9}{4} - x^{2})}} \\ = \frac{1}{2} \int \frac{dx}{\sqrt{(\frac{9}{4} - x^{2})}} \\ = \frac{1}{2} \int \frac{dx}{\sqrt{({(\frac{3}{2})}^{2} - x^{2})}} \end{array}$
We can see that it is the form of $\int \frac{dx}{\sqrt{a^{2} - x^{2}}} = \sin^{- 1} \frac{x}{a} + C$
Hence, replacing $a by \frac{3}{2}$ , we will get
$\begin{array}{l} = \frac{1}{2} \sin^{- 1} \frac{2 x}{3} + C \\ ∴ \int \frac{dx}{\sqrt{9 - 4 x^{2}}} = \frac{1}{2} \sin^{- 1} \frac{2 x}{3} + C \end{array}$
Which is the required answer.