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

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

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integral of sec^3(v)

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

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

Solution steps

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

Apply Integral Reduction

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

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

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

Simplify

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

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

Add a constant to the solution

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

\frac{\partial}{\partial x} (x sec (y))

y^{^{'}} = \frac{cot ( y )}{t}

\int \frac{1}{3} x d x

e^{- 2 x}

(1 - x) y^{^{'}} - 4 x y^{^{'}} + 5 y = cos (x)

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