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

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

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integral of ((cos(θ)-sin(θ)))/(cos^2(θ))

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

ln ∣ tan (θ) + sec (θ) ∣ - sec (θ) + C

Solution steps

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

Expand

\frac{cos ( θ ) - sin ( θ )}{cos ^{2} ( θ )} : \frac{1}{cos ( θ )} - \frac{sin ( θ )}{cos ^{2} ( θ )}

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

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

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

\int \frac{sin ( θ )}{cos ^{2} ( θ )} d θ = sec (θ)

= ln ∣ tan (θ) + sec (θ) ∣ - sec (θ)

Add a constant to the solution

= ln ∣ tan (θ) + sec (θ) ∣ - sec (θ) + C

\frac{\partial}{\partial x} (- 2 y e^{- x})

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

y^{^{''}} - 2 y^{^{'}} - 15 y = - 2

\frac{\partial}{\partial x} (\frac{3 y ^{2} - x ^{2}}{y ^{5}})

\frac{\partial}{\partial y} (e^{y} (x^{2} - y^{2}))

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