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

The integral of (x^3+x-1)/(x+3) is (x^3)/3-(3x^2)/2+10x-31ln|x+3|+3+C

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

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

\frac{x ^{3}}{3} - \frac{3 x ^{2}}{2} + 10 x - 31 ln ∣ x + 3 ∣ + 3 + C

Solution steps

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

Expand

\frac{x ^{3} + x - 1}{x + 3} : \frac{x ^{3}}{x + 3} + \frac{x}{x + 3} - \frac{1}{x + 3}

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{x ^{3}}{x + 3} d x + \int \frac{x}{x + 3} d x - \int \frac{1}{x + 3} d x

\int \frac{x ^{3}}{x + 3} d x = \frac{x ^{3}}{3} - \frac{3 x ^{2}}{2} + 9 x - 27 ln ∣ x + 3 ∣

\int \frac{x}{x + 3} d x = x + 3 - 3 ln ∣ x + 3 ∣

\int \frac{1}{x + 3} d x = ln ∣ x + 3 ∣

= \frac{x ^{3}}{3} - \frac{3 x ^{2}}{2} + 9 x - 27 ln ∣ x + 3 ∣ + x + 3 - 3 ln ∣ x + 3 ∣ - ln ∣ x + 3 ∣

Simplify

= \frac{x ^{3}}{3} - \frac{3 x ^{2}}{2} + 10 x - 31 ln ∣ x + 3 ∣ + 3

Add a constant to the solution

= \frac{x ^{3}}{3} - \frac{3 x ^{2}}{2} + 10 x - 31 ln ∣ x + 3 ∣ + 3 + C

\int csc^{5} (x) d x

\frac{4 x}{( x ^{3} - 3 ) ^{2}}

x + tan (x)

\int \frac{x ^{2}}{x ^{2} + y ^{2}} d y

\int x \frac{1}{1} d x

What is the integral of (x^3+x-1)/(x+3) ?
The integral of (x^3+x-1)/(x+3) is (x^3)/3-(3x^2)/2+10x-31ln|x+3|+3+C