What is the integral of 6sec^6(x) ?

The integral of 6sec^6(x) is 6(tan(x)+(2tan^3(x))/3+(tan^5(x))/5)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 6sec^6(x)

\int 6 sec^{6} (x) d x

6 (tan (x) + \frac{2 tan ^{3} ( x )}{3} + \frac{tan ^{5} ( x )}{5}) + C

Solution steps

\int 6 sec^{6} (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 6 \cdot \int sec^{6} (x) d x

Rewrite using trig identities

= 6 \cdot \int sec^{2} (x) (1 + tan^{2} (x))^{2} d x

Apply u-substitution

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

= 6 \cdot \int (1 + u^{2})^{2} d u

Expand

(1 + u^{2})^{2} : 1 + 2 u^{2} + u^{4}

= 6 \cdot \int 1 + 2 u^{2} + u^{4} 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

= 6 (\int 1 d u + \int 2 u^{2} d u + \int u^{4} d u)

\int 1 d u = u

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

\int u^{4} d u = \frac{u ^{5}}{5}

= 6 (u + \frac{2 u ^{3}}{3} + \frac{u ^{5}}{5})

Substitute back

u = tan (x)

= 6 (tan (x) + \frac{2 tan ^{3} ( x )}{3} + \frac{tan ^{5} ( x )}{5})

Add a constant to the solution

= 6 (tan (x) + \frac{2 tan ^{3} ( x )}{3} + \frac{tan ^{5} ( x )}{5}) + C

h (x) = (\frac{2}{x} - 3) (x^{2} + 7)

x \to 0 lim (ln (\frac{1 + x}{x}))

\int (x^{2} cos (x)) d x

\int sin (\frac{n \cdot π}{3} x) d x

\int \frac{y}{1 + x y} d x

What is the integral of 6sec^6(x) ?
The integral of 6sec^6(x) is 6(tan(x)+(2tan^3(x))/3+(tan^5(x))/5)+C