What is the integral of ((x^2+4))/((x-5)(x^2+9)) ?

The integral of ((x^2+4))/((x-5)(x^2+9)) is

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of ((x^2+4))/((x-5)(x^2+9))

\int \frac{( x ^{2} + 4 )}{( x - 5 ) ( x ^{2} + 9 )} d x

\frac{9}{68} ln x^{2} + 9 + \frac{29}{34} ln ∣ x - 5 ∣ + \frac{15}{34} arctan (\frac{x}{3}) + \frac{1}{51} (- 3 ln x^{2} + 9 - 10 arctan (\frac{x}{3})) + C

Solution steps

\int \frac{x ^{2} + 4}{( x - 5 ) ( x ^{2} + 9 )} d x

Expand

\frac{x ^{2} + 4}{( x - 5 ) ( x ^{2} + 9 )} : \frac{x ^{2}}{( x - 5 ) ( x ^{2} + 9 )} + \frac{4}{( x - 5 ) ( x ^{2} + 9 )}

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 ^{2}}{( x - 5 ) ( x ^{2} + 9 )} d x + \int \frac{4}{( x - 5 ) ( x ^{2} + 9 )} d x

\int \frac{x ^{2}}{( x - 5 ) ( x ^{2} + 9 )} d x = \frac{25}{34} ln ∣ x - 5 ∣ + \frac{1}{34} (\frac{9}{2} ln x^{2} + 9 + 15 arctan (\frac{x}{3}))

\int \frac{4}{( x - 5 ) ( x ^{2} + 9 )} d x = 4 (\frac{1}{34} ln ∣ x - 5 ∣ + \frac{1}{34} (- \frac{1}{2} ln x^{2} + 9 - \frac{5}{3} arctan (\frac{x}{3})))

= \frac{25}{34} ln ∣ x - 5 ∣ + \frac{1}{34} (\frac{9}{2} ln x^{2} + 9 + 15 arctan (\frac{x}{3})) + 4 (\frac{1}{34} ln ∣ x - 5 ∣ + \frac{1}{34} (- \frac{1}{2} ln x^{2} + 9 - \frac{5}{3} arctan (\frac{x}{3})))

Simplify

\frac{25}{34} ln ∣ x - 5 ∣ + \frac{1}{34} (\frac{9}{2} ln x^{2} + 9 + 15 arctan (\frac{x}{3})) + 4 (\frac{1}{34} ln ∣ x - 5 ∣ + \frac{1}{34} (- \frac{1}{2} ln x^{2} + 9 - \frac{5}{3} arctan (\frac{x}{3}))) : \frac{9}{68} ln x^{2} + 9 + \frac{29}{34} ln ∣ x - 5 ∣ + \frac{15}{34} arctan (\frac{x}{3}) + \frac{1}{51} (- 3 ln x^{2} + 9 - 10 arctan (\frac{x}{3}))

= \frac{9}{68} ln x^{2} + 9 + \frac{29}{34} ln ∣ x - 5 ∣ + \frac{15}{34} arctan (\frac{x}{3}) + \frac{1}{51} (- 3 ln x^{2} + 9 - 10 arctan (\frac{x}{3}))

Add a constant to the solution

= \frac{9}{68} ln x^{2} + 9 + \frac{29}{34} ln ∣ x - 5 ∣ + \frac{15}{34} arctan (\frac{x}{3}) + \frac{1}{51} (- 3 ln x^{2} + 9 - 10 arctan (\frac{x}{3})) + C

\int x^{4} e^{(1 - x^{5})} d x

\frac{d y}{d x} + 5 y = 8

\frac{d}{d x} (sin (x^{2} + sin (x^{2} + sin (x^{2}))))

\int 16 x^{2} d x

\frac{d}{d x} (x ∣ x ∣^{∣ x ∣ - 1} (ln ∣ x ∣ + 1))

What is the integral of ((x^2+4))/((x-5)(x^2+9)) ?
The integral of ((x^2+4))/((x-5)(x^2+9)) is