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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

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

Solution steps

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

Expand

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

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

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

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

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

Add a constant to the solution

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

\int (4 x + 1) (2 x^{2} + x - 3)^{4} d x

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

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

\int (x^{5} - 4) d x

\int \frac{1}{3} x ln (- 3 x) d x

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