Question

Perform long division on the​ integrand, write the proper fraction as a sum of partial​ fractions, and then evaluate the integral.
2x3 - 2x + 1/x2 - 2x dx

Answer 1

Answer:

$x^2+4x -\frac{1}{2} lnx + \frac{2}{13} ln(x-2) + C$

Step-by-step explanation:

Given the integrand $\int\limits{\dfrac{2x^3 - 2x + 1}{x^2-2x} } \, dx$, before evaluating the integral function, we will need to simplify the function first by applying long division as shown in the attachment.

Hence the partial form of the function $\dfrac{2x^3 - 2x + 1}{x^2-2x} } = 2x+4 + \frac{6x+1}{x^2-2x}$

Integrating its partial sum

$\int\limits \dfrac{2x^3 - 2x + 1}{x^2-2x} }dx = \int\limits (2x+4 + \frac{6x+1}{x^2-2x})\ dx\\\\= \int\limits {2x} \, dx + \int\limits {4} \, dx + \int\limits {\frac{6x+1}{x^2-2x} \, dx\\ = \frac{2x^2}{2}+4x -\frac{1}{2} \int\limits{\frac{1}{x} } \, dx + \frac{2}{13} \int\limits{\frac{1}{x-2} } \, dx$

$= \frac{2x^2}{2}+4x -\frac{1}{2} lnx + \frac{2}{13} ln(x-2) + C$

$= x^2+4x -\frac{1}{2} lnx + \frac{2}{13} ln(x-2) + C$

NB: Find the partial sum calculation also in the attachment.