What is the integral of (a/(bx)+x/(c^2x+x^3)) ?

The integral of (a/(bx)+x/(c^2x+x^3)) is a/b ln|x|+1/c arctan(x/c)+C

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integral of (a/(bx)+x/(c^2x+x^3))

\int (\frac{a}{b x} + \frac{x}{c ^{2} x + x ^{3}}) d x

\frac{a}{b} ln ∣ x ∣ + \frac{1}{c} arctan (\frac{x}{c}) + C

Solution steps

\int \frac{a}{b x} + \frac{x}{c ^{2} x + x ^{3}} 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{a}{b x} d x + \int \frac{x}{c ^{2} x + x ^{3}} d x

\int \frac{a}{b x} d x = \frac{a}{b} ln ∣ x ∣

\int \frac{x}{c ^{2} x + x ^{3}} d x = \frac{1}{c} arctan (\frac{x}{c})

= \frac{a}{b} ln ∣ x ∣ + \frac{1}{c} arctan (\frac{x}{c})

Add a constant to the solution

= \frac{a}{b} ln ∣ x ∣ + \frac{1}{c} arctan (\frac{x}{c}) + C

\frac{d y}{d x} + 3 y = 0

6 - x^{2}

ln (\frac{1}{3}) (2 + e^{- 3 x})

\int e^{t + 2} d t

What is the integral of (a/(bx)+x/(c^2x+x^3)) ?
The integral of (a/(bx)+x/(c^2x+x^3)) is a/b ln|x|+1/c arctan(x/c)+C