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)

\int sec^{4} (x) d 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

f (x) = 60 + 20 x - 4.5 x^{2}, at x = 2

- x^{2} - x + 9, - x + 7, 0

\int sin^{3} (3 x) cos^{4} (3 x) d x

\frac{\partial}{\partial x} ((1 - x) (1 - y))

x \to 4 + lim (\frac{x - 4}{x ^{2} - 1})

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