What is the integral of (ln(x)+9)/x ?

The integral of (ln(x)+9)/x is (ln^2(x))/2+9ln|x|+C

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integral of (ln(x)+9)/x

\int \frac{ln ( x ) + 9}{x} d x

\frac{ln ^{2} ( x )}{2} + 9 ln ∣ x ∣ + C

Solution steps

\int \frac{ln ( x ) + 9}{x} d x

Expand

\frac{ln ( x ) + 9}{x} : \frac{ln ( x )}{x} + \frac{9}{x}

= \int \frac{ln ( x )}{x} + \frac{9}{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

= \int \frac{ln ( x )}{x} d x + \int \frac{9}{x} d x

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

\int \frac{9}{x} d x = 9 ln ∣ x ∣

= \frac{ln ^{2} ( x )}{2} + 9 ln ∣ x ∣

Add a constant to the solution

= \frac{ln ^{2} ( x )}{2} + 9 ln ∣ x ∣ + C

\frac{\partial}{\partial x} (2 x^{2} + y)

tan (2 (x + 1))

x \to 1 + lim (\frac{x + 2}{1 - x})

\int_{0}^{10} 100 e^{0.1 x} d x

\frac{\partial}{\partial x} (ln (x^{2 + y^{2}}))

What is the integral of (ln(x)+9)/x ?
The integral of (ln(x)+9)/x is (ln^2(x))/2+9ln|x|+C