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

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

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integral of (cos(x)-tan(x))/(cos^2(x))

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

ln ∣ tan (x) + sec (x) ∣ - \frac{tan ^{2} ( x )}{2} + C

Solution steps

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

Expand

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

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

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

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

= ln ∣ tan (x) + sec (x) ∣ - \frac{tan ^{2} ( x )}{2}

Add a constant to the solution

= ln ∣ tan (x) + sec (x) ∣ - \frac{tan ^{2} ( x )}{2} + C

4 x

\frac{d}{d x} (sin (x) + 4 x^{2})

n = 2 \sum \infty \frac{1}{2 i e ^{n} n ^{2}}

\int \frac{1}{e ^{a \cdot x}} d x

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