# How can I integrate rational functions with denominator with just quadratics as factors?

I am told about integration by partial fraction method.I usually guess to decompose a fraction into partial fractions and then I solve the constants.In this problem: $$\int\frac{x^2+2x+6}{(x^2+5x+7)(x^2+6x+7)}$$ I tried to form its partial fraction but I wasn't able to.

Also note that this is an imaginary problem to satisfy my concepts by giving counterpositive statements as equation.My book says:

The rational functions which we shall consider here for integration purposes will be those whose denominators can be factorised into linear and quadratic factors.It is always possible to write the integrand as a sum of simpler rational functions by a method called partial fraction decomposition method.After this, the integration can be carried out easily using the already known methods.

• Partial fractions is the way to go on this, can you show your work so that we can help you find where you got stuck? – Michael Burr Dec 22 '15 at 14:13
• @MichaelBurr I usually use 3 constants unlike the answer so I guess I wasn't able to find out the answer because of it. – Sikander Dec 22 '15 at 14:51
• Whenever you do partial fractions, your numerators should be one degree lower than your denominators (except in the case of powers) – Michael Burr Dec 22 '15 at 18:03

The second factor in the denominator factors into $(x-r_1) (x- r_2)$, where the $r$s are the roots of the quadratic, $$\frac{-6 \pm \sqrt{36-28}}{2}= \frac{-6 \pm \sqrt{8}}{2}= -3 \pm \sqrt{2}$$ So your partial fraction decomp looks like $$\frac{Ax + B}{x^2 + 5x + 7} + \frac{C}{x-r_1} + \frac{D}{x - r_2}$$ You just have to find the right $A, B, C, D$.