What is the integral of (x^5)/(x^4+1) ?

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

Popular >

integral of (x^5)/(x^4+1)

\int \frac{x ^{5}}{x ^{4} + 1} d x

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

Solution steps

\int \frac{x ^{5}}{x ^{4} + 1} d x

Long division

\frac{x ^{5}}{x ^{4} + 1} : x - \frac{x}{x ^{4} + 1}

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

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

\int \frac{x}{x ^{4} + 1} d x = \frac{1}{2} arctan (x^{2})

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

Add a constant to the solution

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

\int z ln (z) d z

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

\int (2 x + 1) \cdot ln (x) d x

\int \frac{( x + 3 ) ^{7}}{2 x} d x

\int 4 y + 2 t - 5 d t

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