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

The integral of (3ln(x)+3)/(2x^2) is (-3ln(x)-6)/(2x)+C

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integral of (3ln(x)+3)/(2x^2)

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

\frac{- 3 ln ( x ) - 6}{2 x} + C

Solution steps

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

Take the constant out:

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

= \frac{1}{2} \cdot \int \frac{3 ln ( x ) + 3}{x ^{2}} d x

Expand

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

= \frac{1}{2} \cdot \int \frac{3 ln ( x )}{x ^{2}} + \frac{3}{x ^{2}} 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}{2} (\int \frac{3 ln ( x )}{x ^{2}} d x + \int \frac{3}{x ^{2}} d x)

\int \frac{3 ln ( x )}{x ^{2}} d x = 3 (- \frac{ln ( x )}{x} - \frac{1}{x})

\int \frac{3}{x ^{2}} d x = - \frac{3}{x}

= \frac{1}{2} (3 (- \frac{ln ( x )}{x} - \frac{1}{x}) - \frac{3}{x})

Simplify

\frac{1}{2} (3 (- \frac{ln ( x )}{x} - \frac{1}{x}) - \frac{3}{x}) : \frac{- 3 ln ( x ) - 6}{2 x}

= \frac{- 3 ln ( x ) - 6}{2 x}

Add a constant to the solution

= \frac{- 3 ln ( x ) - 6}{2 x} + C

What is the integral of (3ln(x)+3)/(2x^2) ?
The integral of (3ln(x)+3)/(2x^2) is (-3ln(x)-6)/(2x)+C