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

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

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

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

ln ∣ tan (x) + sec (x) ∣ + C

Solution steps

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

Rewrite using trig identities

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

Rewrite using trig identities

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

Use the common integral:

\int sec (x) d x = ln ∣ tan (x) + sec (x) ∣

= \frac{1}{2} \cdot 2 ln ∣ tan (x) + sec (x) ∣

Simplify

\frac{1}{2} \cdot 2 ln ∣ tan (x) + sec (x) ∣ : ln ∣ tan (x) + sec (x) ∣

= ln ∣ tan (x) + sec (x) ∣

Add a constant to the solution

= ln ∣ tan (x) + sec (x) ∣ + C

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