What is the integral of (12ln(x))/x ?

The integral of (12ln(x))/x is 6ln^2(x)+C

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integral of (12ln(x))/x

\int \frac{12 ln ( x )}{x} d x

6 ln^{2} (x) + C

Solution steps

\int \frac{12 ln ( x )}{x} d x

Take the constant out:

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

= 12 \cdot \int \frac{ln ( x )}{x} d x

Apply u-substitution

: \int u d u

= 12 \cdot \int u d u

Apply the Power Rule

: \frac{u ^{2}}{2}

= 12 \cdot \frac{u ^{2}}{2}

Substitute back

u = ln (x)

= 12 \cdot \frac{ln ^{2} ( x )}{2}

Simplify

12 \cdot \frac{ln ^{2} ( x )}{2} : 6 ln^{2} (x)

= 6 ln^{2} (x)

Add a constant to the solution

= 6 ln^{2} (x) + C

y = \frac{9}{x}, y = 3 x, y = \frac{1}{3} x

\frac{d}{d x} (e^{- x} (cos (2 x) + sin (2 x)))

e^{- 2 t} cos (3 t) + 5 e^{- 2 t} sin (3 t)

x \to 2 - lim (\frac{x - 2}{x ^{2} - 4})

\frac{x ^{2} - 2}{x ^{2} + 2}