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

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

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

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

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

Solution steps

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

Apply Integration By Parts

: \frac{1}{3} x sec^{3} (x) - \int \frac{1}{3} sec^{3} (x) d x

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

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

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

Add a constant to the solution

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

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