What is the integral of (6ln(x))/x ?

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

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integral of (6ln(x))/x

\int \frac{6 ln ( x )}{x} d x

3 ln^{2} (x) + C

Solution steps

\int \frac{6 ln ( x )}{x} d x

Take the constant out:

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

= 6 \cdot \int \frac{ln ( x )}{x} d x

Apply u-substitution

: \int u d u

= 6 \cdot \int u d u

Apply the Power Rule

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

= 6 \cdot \frac{u ^{2}}{2}

Substitute back

u = ln (x)

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

Simplify

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

= 3 ln^{2} (x)

Add a constant to the solution

= 3 ln^{2} (x) + C

\frac{d}{d x} (\frac{1}{9 + x ^{2}})

\int x^{2} (1 - x)^{5} d x

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

\frac{d}{d t} (- e^{- t^{2}})

y = x^{2}, y = 4, x = 0