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

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

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

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

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

Solution steps

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

Expand

\frac{tan ( x ) + cos ( x )}{sin ( 2 x )} : \frac{tan ( x )}{sin ( 2 x )} + \frac{cos ( x )}{sin ( 2 x )}

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Add a constant to the solution

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

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