What is the integral of (2tan(x))/y ?

The integral of (2tan(x))/y is -2/y ln|cos(x)|+C

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integral of (2tan(x))/y

integral

\frac{2 tan ( x )}{y}

- \frac{2}{y} ln ∣ cos (x) ∣ + C

Solution steps

\int \frac{2 tan ( x )}{y} d x

Take the constant out:

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

= \frac{2}{y} \cdot \int tan (x) d x

Rewrite using trig identities

= \frac{2}{y} \cdot \int \frac{sin ( x )}{cos ( x )} d x

Apply u-substitution

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

= \frac{2}{y} \cdot \int - \frac{1}{u} d u

Take the constant out:

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

= \frac{2}{y} (- \int \frac{1}{u} d u)

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{2}{y} (- ln ∣ u ∣)

Substitute back

u = cos (x)

= \frac{2}{y} (- ln ∣ cos (x) ∣)

Simplify

= - \frac{2}{y} ln ∣ cos (x) ∣

Add a constant to the solution

= - \frac{2}{y} ln ∣ cos (x) ∣ + C

\frac{- 2}{k - 2}

\frac{cos ^{3} ( x )}{( sin ( x ) + cos ( x ) )}

\frac{1}{x ( x + 1 ) ^{\frac{1}{2}}}

\frac{x ^{2}}{( x - 2 ) ( x ^{2} + 1 )}

y = \frac{x ^{2} - cos ( x ) + 1}{x}

What is the integral of (2tan(x))/y ?
The integral of (2tan(x))/y is -2/y ln|cos(x)|+C