What is the integral of xtan(x)sec^4(x) ?

The integral of xtan(x)sec^4(x) is 1/4 xsec^4(x)-1/4 (tan(x)+(tan^3(x))/3)+C

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integral of xtan(x)sec^4(x)

\int x tan (x) sec^{4} (x) d x

\frac{1}{4} x sec^{4} (x) - \frac{1}{4} (tan (x) + \frac{tan ^{3} ( x )}{3}) + C

Solution steps

\int x tan (x) sec^{4} (x) d x

Apply Integration By Parts

: \frac{1}{4} x sec^{4} (x) - \int \frac{1}{4} sec^{4} (x) d x

= \frac{1}{4} x sec^{4} (x) - \int \frac{1}{4} sec^{4} (x) d x

\int \frac{1}{4} sec^{4} (x) d x = \frac{1}{4} (tan (x) + \frac{tan ^{3} ( x )}{3})

= \frac{1}{4} x sec^{4} (x) - \frac{1}{4} (tan (x) + \frac{tan ^{3} ( x )}{3})

Add a constant to the solution

= \frac{1}{4} x sec^{4} (x) - \frac{1}{4} (tan (x) + \frac{tan ^{3} ( x )}{3}) + C

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What is the integral of xtan(x)sec^4(x) ?
The integral of xtan(x)sec^4(x) is 1/4 xsec^4(x)-1/4 (tan(x)+(tan^3(x))/3)+C