What is the integral of (sec^2(x)-1)^2 ?

The integral of (sec^2(x)-1)^2 is 1/3 tan^3(x)+x-tan(x)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (sec^2(x)-1)^2

\int (sec^{2} (x) - 1)^{2} d x

\frac{1}{3} tan^{3} (x) + x - tan (x) + C

Solution steps

\int (sec^{2} (x) - 1)^{2} d x

Rewrite using trig identities

= \int tan^{4} (x) d x

Apply Integral Reduction

= \frac{tan ^{3} ( x )}{3} - \int tan^{2} (x) d x

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

= \frac{tan ^{3} ( x )}{3} - (- x + tan (x))

Simplify

\frac{tan ^{3} ( x )}{3} - (- x + tan (x)) : \frac{1}{3} tan^{3} (x) + x - tan (x)

= \frac{1}{3} tan^{3} (x) + x - tan (x)

Add a constant to the solution

= \frac{1}{3} tan^{3} (x) + x - tan (x) + C

\int 10 (1 + \frac{x}{60})^{2} d x

\int \frac{1}{5 - 4 y ^{2}} d y

\int \frac{1 + x ^{2}}{x 1 + x ^{2}} d x

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

\int \frac{1 - 2 ln ( x )}{x} d x

What is the integral of (sec^2(x)-1)^2 ?
The integral of (sec^2(x)-1)^2 is 1/3 tan^3(x)+x-tan(x)+C