Solve 1 9-25x^2 dx - Vidyadip

MCQ

Solve $\int\frac{1}{\sqrt{9-25\text{x}^2}}\text{dx}$

✓
$\frac{1}{5}\sin^{-1}\big(\frac{5\text{x}}{3}\big)+\text{c}$
B
$\sin^{-1}\big(\frac{5\text{x}}{3}\big)+\text{c}$
C
$\frac{1}{5}\sin^{-1}\big(\frac{3\text{x}}{3}\big)+\text{c}$
D
$\sin^{-1}\big(\frac{3\text{x}}{3}\big)+\text{c}$

✓

Answer

Correct option: A.

$\frac{1}{5}\sin^{-1}\big(\frac{5\text{x}}{3}\big)+\text{c}$

We have,
$\text{I}=\int\frac{\text{dx}}{\sqrt{9-25\text{x}^2}}$
$\text{I}=\int\frac{\text{dx}}{5\sqrt\frac{{9}}{{25}}\text{-x}^2}$
$\text{I}=\frac{1}{5}\int\frac{\text{dx}}{\sqrt{\frac{{3}}{{5}}^2\text{-x}^2}}$
We know that
$\int\frac{\text{dx}}{\text{a}^2-\text{x}^2}=\sin^{-1}\big(\frac{\text{x}}{\text{a}}\big)+\text{C}$
$\therefore\text{I} = \frac{1}{5}\sin^{-1}\Big(\frac{\text{x}}{\frac{{3}}{5}}\Big)+\text{C}$
$\therefore\text{I} = \frac{1}{5}\sin^{-1}\Big(\frac{5\text{x}}{3}\Big)+\text{C}$

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