₀²/3}dx/4+9x² equals - NCERT Class XII Mathematics Chapter -

NCERT Class XII Mathematics Chapter - - Solutions

Choose the correct answer in Exercises 196 and 197.

Question : 197 of 271

Marks: +1, -0

\int\limits_{0}^{2/3}\frac{dx}{4+9x^2}

equals

Solution:

Let

I=\int\limits_{0}^{2/3}\frac{dx}{4+9x^2}

=\frac{1}{9}\int\limits_{0}^{2/3}\frac{dx}{\left(\frac{2}{3}\right)^2 + x^2}

=\frac{1}{9}\times\frac{1}{\frac{2}{3}}\left[\tan^{-1}\left(\frac{3x}{2}\right)\right]_{0}^{2/3}

=\frac{1}{6}\left[\tan^{-1}\left(\frac{3x}{2}\right)\right]_{0}^{2/3}

=\frac{1}{6}\left[\tan^{-1}1-\tan^{-1}0\right]

=\frac{1}{6}\times\frac{\pi}{4}=\frac{\pi}{24}

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