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

The integral of (2+sin^2(x))/(cos^2(x)) is 3tan(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 (2+sin^2(x))/(cos^2(x))

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

3 tan (x) - x + C

Solution steps

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

Expand

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

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

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

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

= 2 tan (x) + tan (x) - x

Simplify

= 3 tan (x) - x

Add a constant to the solution

= 3 tan (x) - x + C

\int \frac{y}{( y ^{2} + 1 )} d y

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

\int \frac{3 x}{x} d x

\int \frac{x}{2 x ( x + 1 )} d x

\int 7 x e^{4 x^{2}} d x

What is the integral of (2+sin^2(x))/(cos^2(x)) ?
The integral of (2+sin^2(x))/(cos^2(x)) is 3tan(x)-x+C