1/9x²+6x+5 - NCERT Class XII Mathematics Chapter -

NCERT Class XII Mathematics Chapter - - Solutions

Exercise - 7.4
Integrate the functions in Exercises 86 to 108.

Question : 96 of 271

Marks: +1, -0

\frac{1}{9x^2+6x+5}

Solution:

Let

I = \int \frac{1}{9x^2+6x+5}dx

= \frac{1}{9} \int \frac{dx}{x^2 + \frac{2}{3}x + \frac{5}{9}}

= \frac{1}{9} \int \frac{dx}{\left(x^2 + \frac{2}{3}x + \frac{1}{9}\right) + \left(\frac{5}{9} - \frac{1}{9}\right)}

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

= \frac{1}{9} \times \frac{1}{\frac{2}{3}} \tan^{-1}\left(\frac{x + \frac{1}{3}}{\frac{2}{3}}\right) + C

= \frac{1}{6} \tan^{-1}\left(\frac{3x+1}{2}\right) + C

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