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

解

\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

解答ステップ

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

部分積分を適用する

= \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) ∣)

定数を解答に追加する

= \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