What is the integral of (x^4+7)/(x^3+9x) ?

The integral of (x^4+7)/(x^3+9x) is (x^2)/2-44/9 ln|x^2+9|+7/9 ln|x|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x^4+7)/(x^3+9x)

\int \frac{x ^{4} + 7}{x ^{3} + 9 x} d x

\frac{x ^{2}}{2} - \frac{44}{9} ln x^{2} + 9 + \frac{7}{9} ln ∣ x ∣ + C

Solution steps

\int \frac{x ^{4} + 7}{x ^{3} + 9 x} d x

Long division

\frac{x ^{4} + 7}{x ^{3} + 9 x} : x + \frac{- 9 x ^{2} + 7}{x ^{3} + 9 x}

= \int x + \frac{- 9 x ^{2} + 7}{x ^{3} + 9 x} 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{- 9 x ^{2} + 7}{x ^{3} + 9 x} d x

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

\int \frac{- 9 x ^{2} + 7}{x ^{3} + 9 x} d x = - \frac{44}{9} ln x^{2} + 9 + \frac{7}{9} ln ∣ x ∣

= \frac{x ^{2}}{2} - \frac{44}{9} ln x^{2} + 9 + \frac{7}{9} ln ∣ x ∣

Add a constant to the solution

= \frac{x ^{2}}{2} - \frac{44}{9} ln x^{2} + 9 + \frac{7}{9} ln ∣ x ∣ + C

\int \frac{2}{25 - x ^{2}} d x

\int cos (12 x) cos (3 x) d x

\int - x^{3} sin (x) d x

\int \frac{x ^{3}}{x ^{2} - 64} d x

\int (csc^{2} (x))^{2} csc (x) d x

What is the integral of (x^4+7)/(x^3+9x) ?
The integral of (x^4+7)/(x^3+9x) is (x^2)/2-44/9 ln|x^2+9|+7/9 ln|x|+C