What is the integral of 6tan^3(x) ?

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

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integral of 6tan^3(x)

\int 6 tan^{3} (x) d x

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

Solution steps

\int 6 tan^{3} (x) d x

Take the constant out:

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

= 6 \cdot \int tan^{3} (x) d x

Rewrite using trig identities

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

Apply u-substitution

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

= 6 \cdot \int \frac{- 1 + u ^{2}}{u} d u

Expand

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

= 6 \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

= 6 (- \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}

= 6 (- ln ∣ u ∣ + \frac{u ^{2}}{2})

Substitute back

u = sec (x)

= 6 (- ln ∣ sec (x) ∣ + \frac{sec ^{2} ( x )}{2})

Add a constant to the solution

= 6 (- ln ∣ sec (x) ∣ + \frac{sec ^{2} ( x )}{2}) + C

45 t^{2}

x \to 0 lim (x \cdot ln (sin (x)))

\frac{d}{d x} (\frac{4 x}{x ^{2} + 6})

y d x - (4 x^{2} y + x) d y = 0

\frac{d}{d x} (ln (x e^{- 7 x}))

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