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

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

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integral of (tan(ln(y)))/y

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

- ln ∣ cos (ln (y)) ∣ + C

Solution steps

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

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 (y)) ∣

Add a constant to the solution

= - ln ∣ cos (ln (y)) ∣ + C

\int 120 x (2 x + 120)^{4} d x

2 x \frac{d y}{d x} = 10 x^{3} y^{5} + y

\frac{\partial}{\partial x} (cos (x - y^{2}))

n = 3 \sum \infty \frac{1}{e n}

\frac{d y}{d x} = 0.0013 y (2500 - y)

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