What is the integral of 1/(3xsec(piln(x))) ?

The integral of 1/(3xsec(piln(x))) is 1/(3pi)sin(piln(x))+C

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integral of 1/(3xsec(piln(x)))

\int \frac{1}{3 x sec ( π ln ( x ) )} d x

\frac{1}{3 π} sin (π ln (x)) + C

Solution steps

\int \frac{1}{3 x sec ( π ln ( x ) )} d x

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{1}{x sec ( π ln ( x ) )} d x

Apply u-substitution

: \int cos (π u) d u

= \frac{1}{3} \cdot \int cos (π u) d u

Apply u-substitution

: \int cos (v) \frac{1}{π} d v

= \frac{1}{3} \cdot \int cos (v) \frac{1}{π} d v

Take the constant out:

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

= \frac{1}{3} \cdot \frac{1}{π} \cdot \int cos (v) d v

Use the common integral:

\int cos (v) d v = sin (v)

= \frac{1}{3} \cdot \frac{1}{π} sin (v)

Substitute back

= \frac{1}{3} \cdot \frac{1}{π} sin (π ln (x))

Simplify

\frac{1}{3} \cdot \frac{1}{π} sin (π ln (x)) : \frac{1}{3 π} sin (π ln (x))

= \frac{1}{3 π} sin (π ln (x))

Add a constant to the solution

= \frac{1}{3 π} sin (π ln (x)) + C

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What is the integral of 1/(3xsec(piln(x))) ?
The integral of 1/(3xsec(piln(x))) is 1/(3pi)sin(piln(x))+C