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Integrate the functions in Exercises 1 to 9.

    Q9.    \sqrt{1 + \frac{x^2}{9}}

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Given function \sqrt{1 + \frac{x^2}{9}},

So, let us consider the function to be;

I = \int\sqrt{1+\frac{x^2}{9}}dx = \frac{1}{3}\int \sqrt{9+x^2}dx

= \frac{1}{3}\int \sqrt{3^2+x^2}dx

And we know that, \int \sqrt{x^2+a^2}dx = \frac{x}{2}\sqrt{x^2+a^2}+\frac{a^2}{2}\log|x+\sqrt{x^2+a^2}| +C

\therefore I = \frac{1}{3}\left [ \frac{x}{2}\sqrt{x^2+9} +\frac{9}{2}\log|x+\sqrt{x^2+9}| \right ]+C

= \frac{x}{6}\sqrt{x^2+9} +\frac{3}{2}\log\left | x+\sqrt{x^2+9} \right |+C

Posted by

Divya Prakash Singh

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