What is the integral of (x^3+1)/(x^3+x^2) ?

The integral of (x^3+1)/(x^3+x^2) is (x^2-x+1)/x-ln|x|-2/x+C

integral of (x^3+1)/(x^3+x^2)

\int \frac{x ^{3} + 1}{x ^{3} + x ^{2}} d x

\frac{x ^{2} - x + 1}{x} - ln ∣ x ∣ - \frac{2}{x} + C

Solution steps

\int \frac{x ^{3} + 1}{x ^{3} + x ^{2}} d x

Apply Integration By Parts

= \frac{x ^{2} - x + 1}{x} - \int \frac{x - 2}{x ^{2}} d x

\int \frac{x - 2}{x ^{2}} d x = ln ∣ x ∣ + \frac{2}{x}

= \frac{x ^{2} - x + 1}{x} - (ln ∣ x ∣ + \frac{2}{x})

Simplify

= \frac{x ^{2} - x + 1}{x} - ln ∣ x ∣ - \frac{2}{x}

Add a constant to the solution

= \frac{x ^{2} - x + 1}{x} - ln ∣ x ∣ - \frac{2}{x} + C

d er i v a t i v e y = e^{- x} arctan (5 e^{x})

d er i v a t i v e x^{2} ln (x^{2} + 1)

im pl i c i t \frac{d y}{d x}, y^{3} = x^{8}

\int \frac{( x + 6 ) ^{4}}{2 x} d x

d er i v a t i v e sin (- 1 (4 x + 1))

What is the integral of (x^3+1)/(x^3+x^2) ?
The integral of (x^3+1)/(x^3+x^2) is (x^2-x+1)/x-ln|x|-2/x+C