What is the integral of 18tan^3(x) ?

The integral of 18tan^3(x) is 18(-ln|sec(x)|+(sec^2(x))/2)+C

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integral of 18tan^3(x)

\int 18 tan^{3} (x) d x

18 (- ln ∣ sec (x) ∣ + \frac{sec ^{2} ( x )}{2}) + C

Solution steps

\int 18 tan^{3} (x) d x

Take the constant out:

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

= 18 \cdot \int tan^{3} (x) d x

Rewrite using trig identities

= 18 \cdot \int (- 1 + sec^{2} (x)) tan (x) d x

Apply u-substitution

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

= 18 \cdot \int \frac{- 1 + u ^{2}}{u} d u

Expand

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

= 18 \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

= 18 (- \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}

= 18 (- ln ∣ u ∣ + \frac{u ^{2}}{2})

Substitute back

u = sec (x)

= 18 (- ln ∣ sec (x) ∣ + \frac{sec ^{2} ( x )}{2})

Add a constant to the solution

= 18 (- ln ∣ sec (x) ∣ + \frac{sec ^{2} ( x )}{2}) + C

(\frac{t ^{3}}{t ^{6} + 9})^{2}

5 e^{x^{2}}

(- 2, 5), (3, 5)

y = e^{8 e^{x}}

\frac{d}{d x} (\frac{1}{8} x^{4} + \frac{1}{4} x^{- 2})

What is the integral of 18tan^3(x) ?
The integral of 18tan^3(x) is 18(-ln|sec(x)|+(sec^2(x))/2)+C