when integrating a fraction where the degree to use long division (or creative thinking) to\nexample…

when integrating a fraction where the degree to use long division (or creative thinking) to\nexample: integrate $\\int \\frac{x^{2}-4 x + 2}{x^{2}+2} d x$\nassigned problems:\n1. $\\int \\frac{x + 1}{x - 1} d x$ 2.

when integrating a fraction where the degree to use long division (or creative thinking) to\nexample: integrate $\\int \\frac{x^{2}-4 x + 2}{x^{2}+2} d x$\nassigned problems:\n1. $\\int \\frac{x + 1}{x - 1} d x$ 2.

Answer

Explanation:

Step1: Simplify the integrand

Use long - division: (\frac{x + 1}{x-1}=\frac{(x - 1)+2}{x - 1}=1+\frac{2}{x - 1})

Step2: Integrate term - by - term

(\int\left(1+\frac{2}{x - 1}\right)dx=\int 1dx+2\int\frac{1}{x - 1}dx)

Step3: Apply the integral formulas

We know that (\int 1dx=x+C_1) and (\int\frac{1}{u}du=\ln|u|+C_2) (where (u=x - 1) and (du=dx))

So, (x + 2\ln|x - 1|+C) (where (C = C_1+2C_2))

Answer:

(x + 2\ln|x - 1|+C)