What is the integral of (x^5+x)/(x+1) ?

The integral of (x^5+x)/(x+1) is (x^5)/5-(x^4)/4+(x^3)/3-(x^2)/2+2x-2ln|x+1|+1+C

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integral of (x^5+x)/(x+1)

\int \frac{x ^{5} + x}{x + 1} d x

\frac{x ^{5}}{5} - \frac{x ^{4}}{4} + \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + 2 x - 2 ln ∣ x + 1 ∣ + 1 + C

Solution steps

\int \frac{x ^{5} + x}{x + 1} d x

Expand

\frac{x ^{5} + x}{x + 1} : \frac{x ^{5}}{x + 1} + \frac{x}{x + 1}

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 ^{5}}{x + 1} d x + \int \frac{x}{x + 1} d x

\int \frac{x ^{5}}{x + 1} d x = \frac{x ^{5}}{5} - \frac{x ^{4}}{4} + \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + x - ln ∣ x + 1 ∣

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

= \frac{x ^{5}}{5} - \frac{x ^{4}}{4} + \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + x - ln ∣ x + 1 ∣ + x + 1 - ln ∣ x + 1 ∣

Simplify

= \frac{x ^{5}}{5} - \frac{x ^{4}}{4} + \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + 2 x - 2 ln ∣ x + 1 ∣ + 1

Add a constant to the solution

= \frac{x ^{5}}{5} - \frac{x ^{4}}{4} + \frac{x ^{3}}{3} - \frac{x ^{2}}{2} + 2 x - 2 ln ∣ x + 1 ∣ + 1 + C

\frac{d}{d x} (\frac{2}{3} (x^{\frac{3}{2}} + y^{\frac{3}{2}}))

e^{t - 1}

f (x) = \frac{- 2 ln ( x ) + 1}{x ^{3}}

ln (1 - 5 y)

What is the integral of (x^5+x)/(x+1) ?
The integral of (x^5+x)/(x+1) is (x^5)/5-(x^4)/4+(x^3)/3-(x^2)/2+2x-2ln|x+1|+1+C