<p>Integration of algebraic fractions, \(\displaystyle\int{\frac{\mathrm{f}\left( x \right)}{\mathrm{g}\left( x \right)}\,\mathrm{d}x}\) and \(\displaystyle\int{\frac{\mathrm{f}(x)}{\sqrt{\mathrm{g}(x)}}\,\mathrm{d}x}\), where&nbsp;\(\mathrm{g}(x)\) is a quadratic polynomial.</p>

For an integral containing \(\dfrac{\mathrm{f}(x)}{\mathrm{g}(x)}\) or \(\dfrac{\mathrm{f}(x)}{\sqrt{\mathrm{g}(x)}}\), inspect the denominator and numerator before choosing a decomposition.

If \(\deg \mathrm{f}\geq\deg \mathrm{g}\), perform polynomial long division first.

If the numerator is constant and \(\mathrm{g}(x)\) is quadratic, complete the square and match an MF27 form.

If \(\mathrm{g}(x)\) factorises, express the proper fraction in partial fractions.

If the earlier routes do not apply, try \(\mathrm{f}(x)=A\mathrm{g}'(x)+B\) and split the integral.

Integration of Algebraic Fractions

Techniques of Integration

International Baccalaureate Mathematics: Analysis and Approaches Higher Level

MF27 denominator form	MF27 denominator form
\(\dfrac{1}{\sqrt{a^2-x^2}}\)	\(\dfrac{1}{a^2+x^2}\)
\(\dfrac{1}{a^2-x^2}\)	\(\dfrac{1}{x^2-a^2}\)

Integrating Algebraic Fractions

Integrating Algebraic Fractions