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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (tan(x))^2

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

tan (x) - x + C

Solution steps

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

Rewrite using trig identities

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

Expand

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

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

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

\int 1 d x = x

= tan (x) - x

Add a constant to the solution

= tan (x) - x + C

y = 2 x^{2} + 4 x - 10

y^{^{''}} + 3 y^{^{'}} + 2 y = 0

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

y y^{^{'}} = 2 x^{3}

y = 78 - 12 cos (x)