INDEFINITE INTEGRAL 1/(1+x^n)
Today´s post we will evaluate an indefinite integral, namely As a special case we will compute (1) By partial fractions we have that where It comes from solving the equation and finding it´s complex roots Without loss of generalty, we will consider n to be odd here, then, one of the roots of the polynomial is , and the other roots will form pairs of complex conjugate roots. To find the coefficients lets do the following The right hand side is just , to evaluate the L.H.S. we apply l`Hopital´s rule to evaluate the limit, But Therefore (2) As mentioned above we have pairs of complex conjugate roots, so if we call the conjugate pair of we may rewrite (1) as (3) Lets focus in one generic pair and the result obtained can be extended to the others (4) Now note that has the form and , so we obtain that and Where To simplify the notation, lets call...