Property 12
\(\displaystyle\int \samefrac{P(x)}{Q(x)}\,dx\)
If \(dP(x)<dQ(x)\) division operation cannot be applied. In this case, expression with multiplication form are converted to addition form.
\(\blacksquare\quad\samefrac{mx+n}{(ax+b)\cdot(cx+d)}=\samefrac A{ax+b}+\samefrac B{cx+d}\)
\(\blacksquare\quad\samefrac{mx^2+nx+p}{(ax^2+bx+c)\cdot(dx+e)}=\samefrac{Ax+B}{ax^2+bx+c}+\samefrac C{dx+e}\)
Example
\(\displaystyle\int \samefrac{3x+4}{x^2+3x+2}\,dx=?\)
Answer
\(\displaystyle\int\samefrac{3x+4}{x^2+3x+2}\,dx=\displaystyle\int\samefrac{3x+4}{(x+1)\cdot(x+2)}\,dx\)
\(\samefrac{3x+4}{(x+1)\cdot(x+2)}=\samefrac A{x+1}+\samefrac B{x+2}\)
\(3x+4=A(x+2)+B(x+1)\)
\(\begin{aligned}x=-2\quad&\Rightarrow3\cdot(-2)+4=A\cdot(2-2)+B\cdot(-2+1)\\&-2=-B\\&2=B\end{aligned}\)
\(\begin{aligned}x=-1\quad&\Rightarrow3\cdot(-1)+4=A\cdot(-1+2)+B\cdot(-1+1)\\&1=A\end{aligned}\)
\(\begin{aligned}\displaystyle\int\samefrac{3x+4}{(x+1)(x+2)}\,dx&=\displaystyle\int\left(\samefrac1{x+1}+\samefrac2{x+2}\right)\,dx\\&=\displaystyle\int\samefrac1{x+1}\,dx+\displaystyle\int\samefrac2{x+2}\,dx\\&=\ln|x+1|+2\ln|x+2|+c\\&=\ln|(x+1)\cdot(x+2)^2|+c\end{aligned}\)
1.
\(\displaystyle\int \samefrac1{x(x+1)}\,dx=?\)
2.
\(\displaystyle\int \samefrac1{x(x+3)}\,dx=?\)
3.
\(\displaystyle\int \samefrac1{x^2-9}\,dx=?\)
4.
\(\displaystyle\int \samefrac1{4x^2-9}\,dx=?\)
5.
\(\displaystyle\int \samefrac1{(2x-1)(2x+3)}\,dx=?\)
6.
\(\displaystyle\int \samefrac{3x+1}{x^2-1}\,dx=?\)
7.
\(\displaystyle\int \samefrac{x-7}{x^2-2x-3}\,dx=?\)