What is the integral of (x^3+4)/(x^5+14x^3+40x) ?

The integral of (x^3+4)/(x^5+14x^3+40x) is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x^3+4)/(x^5+14x^3+40x)

\int \frac{x ^{3} + 4}{x ^{5} + 14 x ^{3} + 40 x} d x

\frac{1}{10} ln ∣ x ∣ + \frac{1}{6} (- \frac{1}{2} ln x^{2} + 4 - 2 arctan (\frac{x}{2})) + \frac{1}{15} (\frac{1}{2} ln x^{2} + 10 + \frac{5 5}{2} arctan (\frac{x}{10})) + C

Solution steps

\int \frac{x ^{3} + 4}{x ^{5} + 14 x ^{3} + 40 x} d x

Take the partial fraction of

\frac{x ^{3} + 4}{x ^{5} + 14 x ^{3} + 40 x} : \frac{1}{10 x} + \frac{- x - 4}{6 ( x ^{2} + 4 )} + \frac{x + 25}{15 ( x ^{2} + 10 )}

= \int \frac{1}{10 x} + \frac{- x - 4}{6 ( x ^{2} + 4 )} + \frac{x + 25}{15 ( x ^{2} + 10 )} 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 \frac{1}{10 x} d x + \int \frac{- x - 4}{6 ( x ^{2} + 4 )} d x + \int \frac{x + 25}{15 ( x ^{2} + 10 )} d x

\int \frac{1}{10 x} d x = \frac{1}{10} ln ∣ x ∣

\int \frac{- x - 4}{6 ( x ^{2} + 4 )} d x = \frac{1}{6} (- \frac{1}{2} ln x^{2} + 4 - 2 arctan (\frac{x}{2}))

\int \frac{x + 25}{15 ( x ^{2} + 10 )} d x = \frac{1}{15} (\frac{1}{2} ln x^{2} + 10 + \frac{5 5}{2} arctan (\frac{x}{10}))

= \frac{1}{10} ln ∣ x ∣ + \frac{1}{6} (- \frac{1}{2} ln x^{2} + 4 - 2 arctan (\frac{x}{2})) + \frac{1}{15} (\frac{1}{2} ln x^{2} + 10 + \frac{5 5}{2} arctan (\frac{x}{10}))

Add a constant to the solution

= \frac{1}{10} ln ∣ x ∣ + \frac{1}{6} (- \frac{1}{2} ln x^{2} + 4 - 2 arctan (\frac{x}{2})) + \frac{1}{15} (\frac{1}{2} ln x^{2} + 10 + \frac{5 5}{2} arctan (\frac{x}{10})) + C

\int \frac{1}{x ^{2} - 8 x + 17} d x

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

\int cos (x) sin (x) x d x

\int (lo g_{10} (x))^{2} d x

\int \frac{ln ( x )}{x} e^{l n^{2} (x)} d x

What is the integral of (x^3+4)/(x^5+14x^3+40x) ?
The integral of (x^3+4)/(x^5+14x^3+40x) is