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.
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 $A, B, C, D$.