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

The integral of (sec^2(ln(x)))/(2x) is 1/2 tan(ln(x))+C

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integral of (sec^2(ln(x)))/(2x)

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

\frac{1}{2} tan (ln (x)) + C

Solution steps

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

Apply u-substitution

= \frac{1}{2} \cdot \int sec^{2} (u) d u

Use the common integral:

\int sec^{2} (u) d u = tan (u)

= \frac{1}{2} tan (u)

Substitute back

u = ln (x)

= \frac{1}{2} tan (ln (x))

Add a constant to the solution

= \frac{1}{2} tan (ln (x)) + C

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What is the integral of (sec^2(ln(x)))/(2x) ?
The integral of (sec^2(ln(x)))/(2x) is 1/2 tan(ln(x))+C