What is the integral of (cos(x))/(sin(2x)) ?

The integral of (cos(x))/(sin(2x)) is 1/2 ln|tan(x/2)|+C

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integral of (cos(x))/(sin(2x))

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

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

Solution steps

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

Rewrite using trig identities

= \int \frac{1}{2 sin ( 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{1}{sin ( x )} d x

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= \frac{1}{2} ln ∣ u ∣

Substitute back

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

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

Add a constant to the solution

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

\frac{1}{x ^{2}}, 2

(2 x^{2} y + 2 1 + x^{4} y^{2}) d x + x^{3} d y = 0

\frac{d}{d x} ((4 x + 3)^{4} (x + 1)^{- 3})

\frac{d q}{d t} + \frac{1}{5 \cdot 1 0 ^{- 9}} q = \frac{200}{1000}

\int \frac{( ln ^{6} ( x ) )}{x} d x

What is the integral of (cos(x))/(sin(2x)) ?
The integral of (cos(x))/(sin(2x)) is 1/2 ln|tan(x/2)|+C