What is the integral of 5x(4/x+3/(x^2)) ?

The integral of 5x(4/x+3/(x^2)) is 5(4x+3ln|x|)+C

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integral of 5x(4/x+3/(x^2))

\int 5 x (\frac{4}{x} + \frac{3}{x ^{2}}) d x

5 (4 x + 3 ln ∣ x ∣) + C

Solution steps

\int 5 x (\frac{4}{x} + \frac{3}{x ^{2}}) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 5 \cdot \int x (\frac{4}{x} + \frac{3}{x ^{2}}) d x

Simplify

x (\frac{4}{x} + \frac{3}{x ^{2}}) : 4 + \frac{3}{x}

= 5 \cdot \int 4 + \frac{3}{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

= 5 (\int 4 d x + \int \frac{3}{x} d x)

\int 4 d x = 4 x

\int \frac{3}{x} d x = 3 ln ∣ x ∣

= 5 (4 x + 3 ln ∣ x ∣)

Add a constant to the solution

= 5 (4 x + 3 ln ∣ x ∣) + C

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What is the integral of 5x(4/x+3/(x^2)) ?
The integral of 5x(4/x+3/(x^2)) is 5(4x+3ln|x|)+C