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

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

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integral of sec^3(x)cot(x)

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

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

Solution steps

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

Rewrite using trig identities

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

Apply Integration By Parts

: sec (x) - \int - cot (x) csc (x) tan (x) d x

= sec (x) - \int - cot (x) csc (x) tan (x) d x

\int - cot (x) csc (x) tan (x) d x = - ln tan (\frac{x}{2})

= sec (x) - (- ln tan (\frac{x}{2}))

Simplify

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

Add a constant to the solution

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

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