What is the integral of (x^2-x+42)/(x^3+7x) ?

The integral of (x^2-x+42)/(x^3+7x) is 6ln|x|-5/2 ln|x^2+7|-1/(sqrt(7))arctan(x/(sqrt(7)))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (x^2-x+42)/(x^3+7x)

\int \frac{x ^{2} - x + 42}{x ^{3} + 7 x} d x

6 ln ∣ x ∣ - \frac{5}{2} ln x^{2} + 7 - \frac{1}{7} arctan (\frac{x}{7}) + C

Solution steps

\int \frac{x ^{2} - x + 42}{x ^{3} + 7 x} d x

Take the partial fraction of

\frac{x ^{2} - x + 42}{x ^{3} + 7 x} : \frac{6}{x} + \frac{- 5 x - 1}{x ^{2} + 7}

= \int \frac{6}{x} + \frac{- 5 x - 1}{x ^{2} + 7} 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{6}{x} d x + \int \frac{- 5 x - 1}{x ^{2} + 7} d x

\int \frac{6}{x} d x = 6 ln ∣ x ∣

\int \frac{- 5 x - 1}{x ^{2} + 7} d x = - \frac{5}{2} ln x^{2} + 7 - \frac{1}{7} arctan (\frac{x}{7})

= 6 ln ∣ x ∣ - \frac{5}{2} ln x^{2} + 7 - \frac{1}{7} arctan (\frac{x}{7})

Add a constant to the solution

= 6 ln ∣ x ∣ - \frac{5}{2} ln x^{2} + 7 - \frac{1}{7} arctan (\frac{x}{7}) + C

y = x^{\frac{1}{2}}

f (x) = 2 x + 5, at x = 2

n = 0 \sum \infty \frac{10 0 ^{n}}{n !}

\frac{1 - 2 s}{s ^{2} + 4 s + 5}

\frac{d}{d x} (2 sin (8 x) cos (x))

What is the integral of (x^2-x+42)/(x^3+7x) ?
The integral of (x^2-x+42)/(x^3+7x) is 6ln|x|-5/2 ln|x^2+7|-1/(sqrt(7))arctan(x/(sqrt(7)))+C