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

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

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

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

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

Solution steps

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

Rewrite using trig identities

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

Rewrite using trig identities

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

Apply Integration By Parts

: sec (x) - \int - cot (x) csc (x) tan (x) d x

= \frac{1}{2} (sec (x) - \int - cot (x) csc (x) tan (x) d x)

\int - cot (x) csc (x) tan (x) d x = - ln tan (\frac{x}{2})

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

Simplify

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

Add a constant to the solution

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

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