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

The integral of (tan(ln(x)))/x is -ln|cos(ln(x))|+C

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integral of (tan(ln(x)))/x

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

- ln ∣ cos (ln (x)) ∣ + C

Solution steps

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

Apply u-substitution

: \int tan (u) d u

= \int tan (u) d u

Rewrite using trig identities

= \int \frac{sin ( u )}{cos ( u )} d u

Apply u-substitution

: \int - \frac{1}{v} d v

= \int - \frac{1}{v} d v

Take the constant out:

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

= - \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= - ln ∣ v ∣

Substitute back

= - ln ∣ cos (ln (x)) ∣

Add a constant to the solution

= - ln ∣ cos (ln (x)) ∣ + C

(3 x^{3} + x^{- 3}) (x + 3) (x^{2} - 5)

f^{^{'}} (x) = 2 x^{3} - x

y = c e^{\frac{- x ^{3}}{3}} + 1

\int_{1}^{5} 6250 π x d x

f (x) = x^{3} + x

What is the integral of (tan(ln(x)))/x ?
The integral of (tan(ln(x)))/x is -ln|cos(ln(x))|+C