What is the integral of (x^4)/((x+1)) ?

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

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integral of (x^4)/((x+1))

integral

\frac{x ^{4}}{( x + 1 )}

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

Solution steps

\int \frac{x ^{4}}{x + 1} d x

Long division

\frac{x ^{4}}{x + 1} : x^{3} - x^{2} + x - 1 + \frac{1}{x + 1}

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

Apply the Sum Rule:

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

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

\int x^{3} d x = \frac{x ^{4}}{4}

\int x^{2} d x = \frac{x ^{3}}{3}

\int x d x = \frac{x ^{2}}{2}

\int 1 d x = x

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

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

Add a constant to the solution

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

\frac{( x ^{2} - x - 2 )}{( 1 - x ^{2} )}

\frac{x ^{\frac{1}{2}}}{( x ^{2} + 1 ) ^{2}}

\frac{x}{( x ^{2} + 9 )}

x (x + 4)^{8}

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

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