Question

Integrate the following function​

Answer 1

Step-by-step explanation:

We have,

$\int \frac{ {x}^{3} + 3 {x}^{2} + 2x + 1 }{x - 1} dx$

$= \int \frac{ {x}^{3} - {x}^{2} + 4 {x}^{2} - 4x + 6x - 6 + 5 }{x - 1} dx$

$= \int \frac{ {x}^{2}(x - 1) + 4x(x - 1) + 6(x - 1) + 5 }{x - 1} dx$

$= \int \bigg[ \frac{ {x}^{2}(x - 1)}{x - 1} + \frac{4x(x - 1) }{x - 1} + \frac{ 6(x - 1)}{x - 1} + \frac{ 5 }{x - 1} \bigg] dx$

$= \int \frac{ {x}^{2}(x - 1)}{x - 1}dx + \int \frac{4x(x - 1) }{x - 1} dx+ \int \frac{ 6(x - 1)}{x - 1} dx+ \int \frac{ 5 }{x - 1} dx$

$= \int {x}^{2}dx + \int 4xdx+ \int 6dx+5 \int \frac{ 1 }{x - 1} dx$

$= \frac{{x}^{3}}{3}+ 4 .\frac{ {x}^{2} }{2} + 6x+5 \ln(x - 1) + C$

$= \frac{{x}^{3}}{3}+ 2 {x}^{2} + 6x+5 \ln(x - 1) + C$