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

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

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integral of [sec(x)+sec^3(x)]

\int [sec (x) + sec^{3} (x)] d x

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

Solution steps

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Simplify

ln ∣ tan (x) + sec (x) ∣ + \frac{1}{2} sec (x) tan (x) + \frac{1}{2} ln ∣ tan (x) + sec (x) ∣ : \frac{1}{2} sec (x) tan (x) + \frac{3}{2} ln ∣ tan (x) + sec (x) ∣

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

Add a constant to the solution

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

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