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

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

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integral of 2sec^3(x)

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

sec (x) tan (x) + ln ∣ tan (x) + sec (x) ∣ + C

Solution steps

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

Take the constant out:

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

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

Apply Integral Reduction

= 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) ∣

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

Simplify

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

= sec (x) tan (x) + ln ∣ tan (x) + sec (x) ∣

Add a constant to the solution

= sec (x) tan (x) + ln ∣ tan (x) + sec (x) ∣ + C

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