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

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

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

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

- ln ∣ tan (x) + sec (x) ∣ + 2 sin (x) + C

Solution steps

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

Rewrite using trig identities

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

Expand

\frac{- 1 + 2 cos ^{2} ( x )}{cos ( x )} : - \frac{1}{cos ( x )} + 2 cos (x)

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

\int \frac{1}{cos ( x )} d x = ln ∣ tan (x) + sec (x) ∣

\int 2 cos (x) d x = 2 sin (x)

= - ln ∣ tan (x) + sec (x) ∣ + 2 sin (x)

Add a constant to the solution

= - ln ∣ tan (x) + sec (x) ∣ + 2 sin (x) + C

8 x^{2} y^{^{'}} = y^{^{'}} + 8 x e^{- y}

f (x) = - 7 x - \frac{2}{x ^{7}}

\frac{\partial}{\partial x} (arctan (y))

x \to 2 lim (π)

\frac{d}{d x} (- sin (x) + cos (x))

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