What is the integral of ((x+5))/((-3x)) ?

The integral of ((x+5))/((-3x)) is -1/3 (x+5ln|x|)+C

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integral of ((x+5))/((-3x))

integral

\frac{( x + 5 )}{( - 3 x )}

- \frac{1}{3} (x + 5 ln ∣ x ∣) + C

Solution steps

\int \frac{x + 5}{- 3 x} d x

Take the constant out:

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

= \frac{1}{- 3} \cdot \int \frac{x + 5}{x} d x

Simplify

= (- \frac{1}{3}) \cdot \int \frac{x + 5}{x} d x

Expand

\frac{x + 5}{x} : 1 + \frac{5}{x}

= (- \frac{1}{3}) \cdot \int 1 + \frac{5}{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

= (- \frac{1}{3}) (\int 1 d x + \int \frac{5}{x} d x)

\int 1 d x = x

\int \frac{5}{x} d x = 5 ln ∣ x ∣

= (- \frac{1}{3}) (x + 5 ln ∣ x ∣)

Simplify

= - \frac{1}{3} (x + 5 ln ∣ x ∣)

Add a constant to the solution

= - \frac{1}{3} (x + 5 ln ∣ x ∣) + C

sin^{3} (x) - 1

\frac{x ^{3} - x + 1}{x ^{3}}

\frac{x ^{2} ln ( x )}{e ^{x}}

8 x^{- 1}

\frac{1}{( cos ( 2 x ) )}

What is the integral of ((x+5))/((-3x)) ?
The integral of ((x+5))/((-3x)) is -1/3 (x+5ln|x|)+C