What is the integral of (sec^3(x))/2 ?

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

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integral of (sec^3(x))/2

\int \frac{sec ^{3} ( x )}{2} d x

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

Solution steps

\int \frac{sec ^{3} ( x )}{2} d x

Take the constant out:

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

= \frac{1}{2} \cdot \int sec^{3} (x) d x

Apply Integral Reduction

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

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

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

Simplify

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

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

Add a constant to the solution

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

\frac{\partial}{\partial x} (\frac{1}{2} (x^{2} - y^{2}))

\frac{d}{d t} (2 sin^{2} (t))

x \to 0 lim (x^{2} + 2 x)

\int x^{- 0.5} d x

\frac{d}{d x} (0.0225 x e^{- 0.0467 x})

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