What is the integral of 1/x n(x) ?

The integral of 1/x n(x) is d/2 ln^2(x)+C

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integral of 1/x n(x)

\int \frac{1}{x} d ln (x) d x

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

Solution steps

\int \frac{1}{x} d ln (x) d x

Take the constant out:

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

= d \cdot \int \frac{1}{x} ln (x) d x

Apply u-substitution

: \int u d u

= d \cdot \int u d u

Apply the Power Rule

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

= d \frac{u ^{2}}{2}

Substitute back

u = ln (x)

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

Simplify

d \frac{ln ^{2} ( x )}{2} : \frac{d}{2} ln^{2} (x)

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

Add a constant to the solution

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

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