What is the integral of 3sec^6(t) ?

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 3sec^6(t)

\int 3 sec^{6} (t) d t

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

Solution steps

\int 3 sec^{6} (t) d t

Take the constant out:

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

= 3 \cdot \int sec^{6} (t) d t

Rewrite using trig identities

= 3 \cdot \int sec^{2} (t) (1 + tan^{2} (t))^{2} d t

Apply u-substitution

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

= 3 \cdot \int (1 + u^{2})^{2} d u

Expand

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

= 3 \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

= 3 (\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}

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

Substitute back

u = tan (t)

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

Simplify

3 (tan (t) + \frac{2 tan ^{3} ( t )}{3} + \frac{tan ^{5} ( t )}{5}) : 3 tan (t) + 2 tan^{3} (t) + \frac{3}{5} tan^{5} (t)

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

Add a constant to the solution

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

\frac{\partial}{\partial x} (3 cos^{2} (x))

\int lo g_{8} (3 x) d x

\int_{1}^{9} x d x

\int (\frac{1}{cos ( 4 x )})^{2} d x

y^{'} = 0.0012 y (1000 - y)

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