What is the integral of sec^3(3x) ?

The integral of sec^3(3x) is 1/3 ((sec^2(3x)sin(3x))/2+1/2 ln|tan(3x)+sec(3x)|)+C

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integral of sec^3(3x)

\int sec^{3} (3 x) d x

\frac{1}{3} (\frac{sec ^{2} ( 3 x ) sin ( 3 x )}{2} + \frac{1}{2} ln ∣ tan (3 x) + sec (3 x) ∣) + C

Solution steps

\int sec^{3} (3 x) d x

Apply u-substitution

: \int sec^{3} (u) \frac{1}{3} d u

= \int sec^{3} (u) \frac{1}{3} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int sec^{3} (u) d u

Apply Integral Reduction

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

\int sec (u) d u = ln ∣ tan (u) + sec (u) ∣

= \frac{1}{3} (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

Substitute back

u = 3 x

= \frac{1}{3} (\frac{sec ^{2} ( 3 x ) sin ( 3 x )}{2} + \frac{1}{2} ln ∣ tan (3 x) + sec (3 x) ∣)

Add a constant to the solution

= \frac{1}{3} (\frac{sec ^{2} ( 3 x ) sin ( 3 x )}{2} + \frac{1}{2} ln ∣ tan (3 x) + sec (3 x) ∣) + C

\frac{d}{d x} (1 0^{3 x})

\frac{\partial}{\partial x} (\frac{x ^{2}}{y})

x \to 2 lim (x^{2} - 3)

6 x^{2} y^{^{'}} = y^{^{'}} + 2 x e^{- y}

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

What is the integral of sec^3(3x) ?
The integral of sec^3(3x) is 1/3 ((sec^2(3x)sin(3x))/2+1/2 ln|tan(3x)+sec(3x)|)+C