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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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