What is the integral of sec^4(x) ?

The integral of sec^4(x) is tan(x)+(tan^3(x))/3+C

Popular >

integral of sec^4(x)

integral

sec^{4} (x)

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

Solution steps

\int sec^{4} (x) d x

Rewrite using trig identities

= \int (1 + tan^{2} (x)) sec^{2} (x) d x

Apply u-substitution

: \int 1 + u^{2} d u

= \int 1 + u^{2} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 1 d u + \int u^{2} d u

\int 1 d u = u

\int u^{2} d u = \frac{u ^{3}}{3}

= u + \frac{u ^{3}}{3}

Substitute back

u = tan (x)

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

Add a constant to the solution

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

\int \frac{4 - x}{2 + x} d x

\int x^{\frac{- 5}{4}} d x

\int \frac{18 arctan ( 2 x )}{1 + 4 x ^{2}} d x

\int cos (x) (- sin (x)) d x

\int cos^{3} (3 x) 3 sin (3 x) d x

What is the integral of sec^4(x) ?
The integral of sec^4(x) is tan(x)+(tan^3(x))/3+C