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

The integral of (cos(2x))/(cos(x)-sin(x)) is sin(x)-cos(x)+C

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

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

sin (x) - cos (x) + C

Solution steps

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

\frac{cos ( 2 x )}{cos ( x ) - sin ( x )} = \frac{( cos ( 2 x ) ) ( cos ( x ) + sin ( x ) )}{( cos ( x ) ) ^{2} - ( sin ( x ) ) ^{2}}

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

Rewrite using trig identities

= \int cos (x) + sin (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 cos (x) d x + \int sin (x) d x

\int cos (x) d x = sin (x)

\int sin (x) d x = - cos (x)

= sin (x) - cos (x)

Add a constant to the solution

= sin (x) - cos (x) + C

\frac{d}{d x} (5 cos^{- 4} (x))

\int \frac{e ^{2 x}}{e ^{2 x} - 1} d x

\frac{d}{d x} (sin (x + π))

x^{2} e^{x}

n = 0 \sum \infty tan (n)

What is the integral of (cos(2x))/(cos(x)-sin(x)) ?
The integral of (cos(2x))/(cos(x)-sin(x)) is sin(x)-cos(x)+C