What is the integral of 1/(2sin(x)cos(x)) ?

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

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integral of 1/(2sin(x)cos(x))

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

\frac{1}{2} ln ∣ tan (x) ∣ + C

Solution steps

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

Rewrite using trig identities

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

Apply Trigonometric 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 (x)

= \frac{1}{2} ln ∣ tan (x) ∣

Add a constant to the solution

= \frac{1}{2} ln ∣ tan (x) ∣ + C

\frac{d}{d x} (2 x ln (x) + x)

1500 x + 3000 (3 x + 3)^{- 1} - 1000

x^{\frac{2}{7}}, 1, x = 0

8 x y

(2 x + \frac{2}{y}) + (\frac{2 x ^{2}}{y} + \frac{2 x}{y ^{2}}) y^{^{'}} = 0

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