Property 11
Simple Fractionization
\(P(x),Q(x)\quad\text{(polynomial)}\)
\(\displaystyle\int \samefrac{P(x)}{Q(x)}\,dx\)
\(dP(x)\ge dQ(x)\)
If \(dp(x)-dQ(x)\), by long division it can be fractionized.
Example
\(\displaystyle\int \left(\samefrac{x^3+3x+7}x\right)\,dx=?\)
Answer
\(\begin{aligned}\displaystyle\int\left(\samefrac{x^3+3x+7}x\right)\,dx&=\displaystyle\int\left(\samefrac{x^3}x+\samefrac{3x}x+\samefrac7x\right)\,dx\\&=\displaystyle\int\left(x^2+3+\samefrac7x\right)\,dx\\&=\samefrac{x^3}3+3x+7\cdot\ln|x|+c\end{aligned}\)
1.
\(\displaystyle\int \samefrac{x^2+4x}x\,dx=?\)
2.
\(\displaystyle\int \samefrac{x^3-1}{2x}\,dx=?\)
3.
\(\displaystyle\int \samefrac{4x+2}{x+1}\,dx=?\)
4.
\(\displaystyle\int \samefrac{x^2-3x+1}{x-1}\,dx=?\)
5.
\(\displaystyle\int \samefrac{x^2+2x+7}{x-3}\,dx=?\)
6.
\(\displaystyle\int \samefrac{x^2-x+3}{x+1}\,dx=?\)
7.
\(\displaystyle\int \samefrac{x^2+4}{x^2+1}\,dx=?\)
8.
\(\displaystyle\int \samefrac{3x+1}{2x-2}\,dx=?\)
9.
\(\displaystyle\int \samefrac{2x^2-x+5}{x+1}\,dx=?\)
10.
\(\displaystyle\int \samefrac{5x+1}{2x-1}\,dx=?\)