Integrate 5(x + 2)/(x + 1)(x + 6) with respect to x

This term cannot be integrated in the form that it is in. We will have to do some algebraic manipulation to rewrite it. The term can be split into partial fractions because the denominator of the term has been factorised. There are two brackets in the denominator so we can split the fraction into two partial fractions (whiteboard sketch). 5(x+2)/(x+1)(x+6)= A/(x+1) + B/(x+6), we can multiply through by the denominators (whiteboard sketch) and end up with 5(x+2)=A(x+6) + B(x+1). Now we need to work out what A and B are. This will be done by substitution. We can see that if we enter x=-6, the A term will disappear and if we enter x=-1 then the B term disappears. We can use x=-6 to work out that B=4 (show sketch) and when we use x=-1 we can see A=1. We now need to integrate 1/(x+1) and 4/(x+6). This uses the standard rule we learn in C4, where we end up with the natural logarithm of the denominator, preceded by the coefficient on the numerator (whiteboard sketch). The final answer is ln(x+1) + 4ln(x+6) + c