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  • Integrate the functions in Exercises 1 to 24. 1

Integrate the functions in Exercises 1 to 24.

    Q1.    \frac{1}{x - x^3}

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Firstly we will simplify  the given equation :-

                     \frac{1}{x - x^3}\ =\ \frac{1}{(x)(1-x)(1+x)}

Let                         

                                      \frac{1}{(x)(1-x)(1+x)} =\ \frac{A}{x}\ +\ \frac{B}{1-x}\ +\ \frac{C}{1+x}     

By solving the equation and equating the coefficients of x2, x and the constant term, we get

                                   A\ =\ 1,\ B\ =\ \frac{1}{2},\ C\ =\ \frac{-1}{2}

Thus the integral can be written as :

                                    \int \frac{1}{(x)(1-x)(1+x)}dx =\ \int \frac{1}{x}dx\ +\ \frac{1}{2}\int \frac{1}{1-x}dx\ +\ \frac{-1}{2}\int \frac{1}{1+x}dx

                                                                                       =\ \log x\ -\ \frac{1}{2}\log(1-x)\ +\ \frac{-1}{2}\log (1+x)

or                                                                                    =\ \frac{1}{2} \log \frac{x^2}{1-x^2}\ +\ C

Posted by

Devendra Khairwa

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