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

The integral of (2tan(x)-cot(x))^2 is -9x+4tan(x)-cot(x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (2tan(x)-cot(x))^2

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

- 9 x + 4 tan (x) - cot (x) + C

Solution steps

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

Rewrite using trig identities

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

Expand

\frac{( 2 sin ^{2} ( x ) - cos ^{2} ( x ) ) ^{2}}{cos ^{2} ( x ) sin ^{2} ( x )} : \frac{4 sin ^{2} ( x )}{cos ^{2} ( x )} - 4 + \frac{cos ^{2} ( x )}{sin ^{2} ( x )}

= \int \frac{4 sin ^{2} ( x )}{cos ^{2} ( x )} - 4 + \frac{cos ^{2} ( x )}{sin ^{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{4 sin ^{2} ( x )}{cos ^{2} ( x )} d x - \int 4 d x + \int \frac{cos ^{2} ( x )}{sin ^{2} ( x )} d x

\int \frac{4 sin ^{2} ( x )}{cos ^{2} ( x )} d x = 4 (tan (x) - x)

\int 4 d x = 4 x

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

= 4 (tan (x) - x) - 4 x - cot (x) - x

Simplify

4 (tan (x) - x) - 4 x - cot (x) - x : - 9 x + 4 tan (x) - cot (x)

= - 9 x + 4 tan (x) - cot (x)

Add a constant to the solution

= - 9 x + 4 tan (x) - cot (x) + C

\int cot (25 x) d x

\int x sinh (2 - x^{2}) d x

\int 2 x^{2} e^{- x^{2}} d x

\int \frac{1 + e ^{x}}{x + e ^{x}} d x

\int 8 x + 3 d x

What is the integral of (2tan(x)-cot(x))^2 ?
The integral of (2tan(x)-cot(x))^2 is -9x+4tan(x)-cot(x)+C