What is the integral of 3tan^3(θ) ?

The integral of 3tan^3(θ) is 3(-ln|sec(θ)|+(sec^2(θ))/2)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 3tan^3(θ)

\int 3 tan^{3} (θ) d θ

3 (- ln ∣ sec (θ) ∣ + \frac{sec ^{2} ( θ )}{2}) + C

Solution steps

\int 3 tan^{3} (θ) d θ

Take the constant out:

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

= 3 \cdot \int tan^{3} (θ) d θ

Rewrite using trig identities

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

Apply u-substitution

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

= 3 \cdot \int \frac{- 1 + u ^{2}}{u} d u

Expand

\frac{- 1 + u ^{2}}{u} : - \frac{1}{u} + u

= 3 \cdot \int - \frac{1}{u} + u 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 \frac{1}{u} d u + \int u d u)

\int \frac{1}{u} d u = ln ∣ u ∣

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

= 3 (- ln ∣ u ∣ + \frac{u ^{2}}{2})

Substitute back

u = sec (θ)

= 3 (- ln ∣ sec (θ) ∣ + \frac{sec ^{2} ( θ )}{2})

Add a constant to the solution

= 3 (- ln ∣ sec (θ) ∣ + \frac{sec ^{2} ( θ )}{2}) + C

y^{^{'}} = x y^{2} (1 + x^{2})^{- \frac{1}{2}}, y (0) = 2

(4, 1), (6, 7)

\frac{\partial}{\partial x} ((1 + \frac{2}{y}) cos (x))

\frac{d}{d x} (3 cos (x^{2} - 1))

\int_{5}^{7} \frac{1}{4 + ( x - 5 ) ^{2}} d x

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