What is the integral of (x^3+93x+70)/(x^3+100x) ?

The integral of (x^3+93x+70)/(x^3+100x) is x+7/10 ln|x|-7/20 ln|x^2+100|-7/10 arctan(x/(10))+C

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integral of (x^3+93x+70)/(x^3+100x)

\int \frac{x ^{3} + 93 x + 70}{x ^{3} + 100 x} d x

x + \frac{7}{10} ln ∣ x ∣ - \frac{7}{20} ln x^{2} + 100 - \frac{7}{10} arctan (\frac{x}{10}) + C

Solution steps

\int \frac{x ^{3} + 93 x + 70}{x ^{3} + 100 x} d x

Expand

\frac{x ^{3} + 93 x + 70}{x ^{3} + 100 x} : \frac{x ^{3}}{x ^{3} + 100 x} + \frac{93 x}{x ^{3} + 100 x} + \frac{70}{x ^{3} + 100 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{x ^{3}}{x ^{3} + 100 x} d x + \int \frac{93 x}{x ^{3} + 100 x} d x + \int \frac{70}{x ^{3} + 100 x} d x

\int \frac{x ^{3}}{x ^{3} + 100 x} d x = - 10 arctan (\frac{x}{10}) + x

\int \frac{93 x}{x ^{3} + 100 x} d x = \frac{93}{10} arctan (\frac{x}{10})

\int \frac{70}{x ^{3} + 100 x} d x = 70 (\frac{1}{100} ln ∣ x ∣ - \frac{1}{200} ln x^{2} + 100)

= - 10 arctan (\frac{x}{10}) + x + \frac{93}{10} arctan (\frac{x}{10}) + 70 (\frac{1}{100} ln ∣ x ∣ - \frac{1}{200} ln x^{2} + 100)

Simplify

- 10 arctan (\frac{x}{10}) + x + \frac{93}{10} arctan (\frac{x}{10}) + 70 (\frac{1}{100} ln ∣ x ∣ - \frac{1}{200} ln x^{2} + 100) : x + \frac{7}{10} ln ∣ x ∣ - \frac{7}{20} ln x^{2} + 100 - \frac{7}{10} arctan (\frac{x}{10})

= x + \frac{7}{10} ln ∣ x ∣ - \frac{7}{20} ln x^{2} + 100 - \frac{7}{10} arctan (\frac{x}{10})

Add a constant to the solution

= x + \frac{7}{10} ln ∣ x ∣ - \frac{7}{20} ln x^{2} + 100 - \frac{7}{10} arctan (\frac{x}{10}) + C

y = x^{3} - \frac{x ^{2}}{16} + 4 x + 5

f (x) = 2 x^{2} + 3 x - 2, at x = - 1

\frac{d}{d x} (k (1 - \frac{w}{d} e^{- s x} sin (d x + t)))

- x

\frac{9 - x ^{2}}{16 x}

What is the integral of (x^3+93x+70)/(x^3+100x) ?
The integral of (x^3+93x+70)/(x^3+100x) is x+7/10 ln|x|-7/20 ln|x^2+100|-7/10 arctan(x/(10))+C