What is the integral of (sec(x))^3 ?

The integral of (sec(x))^3 is -xsec^3(x)cos^3(x)+x+1/2 sec(x)tan(x)+1/2 ln|tan(x)+sec(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(x))^3

\int (sec (x))^{3} d x

- x sec^{3} (x) cos^{3} (x) + x + \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣ + C

Solution steps

\int (sec (x))^{3} d x

Rewrite using trig identities

= \int \frac{1}{cos ^{3} ( x )} d x

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

= \int sec^{3} (x) (1 - cos^{3} (x)) + 1 d x

Rewrite using trig identities

= \int \frac{1 - cos ^{3} ( x )}{cos ^{3} ( x )} + 1 d x

Apply Integration By Parts

: x (sec^{3} (x) (1 - cos^{3} (x)) + 1) - \int 3 x sec^{4} (x) sin (x) d x

= x (sec^{3} (x) (1 - cos^{3} (x)) + 1) - \int 3 x sec^{4} (x) sin (x) d x

\int 3 x sec^{4} (x) sin (x) d x = x sec^{3} (x) - \frac{1}{2} sec (x) tan (x) - \frac{1}{2} ln ∣ tan (x) + sec (x) ∣

= x (sec^{3} (x) (1 - cos^{3} (x)) + 1) - (x sec^{3} (x) - \frac{1}{2} sec (x) tan (x) - \frac{1}{2} ln ∣ tan (x) + sec (x) ∣)

Simplify

x (sec^{3} (x) (1 - cos^{3} (x)) + 1) - (x sec^{3} (x) - \frac{1}{2} sec (x) tan (x) - \frac{1}{2} ln ∣ tan (x) + sec (x) ∣) : - x sec^{3} (x) cos^{3} (x) + x + \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣

= - x sec^{3} (x) cos^{3} (x) + x + \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣

Add a constant to the solution

= - x sec^{3} (x) cos^{3} (x) + x + \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣ + C

\frac{7 x}{2 ^{x}}

(- 8, - 4), (- 6, - 1)

y = \frac{4}{3} x + 2

x \to \infty lim (\frac{+ x - 2 ^{+ x}}{3 ^{x} - 5})

\int 5 \frac{cos ( x ) + sin ( x )}{sin ( 2 x )} d x

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