integral of (sec^3(x))/2

解

\int \frac{sec ^{3} ( x )}{2} d x

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

解答ステップ

\int \frac{sec ^{3} ( x )}{2} d x

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int sec^{3} (x) d x

積分換算を適用する

= \frac{1}{2} (\frac{sec ^{2} ( x ) sin ( x )}{2} + \frac{1}{2} \cdot \int sec (x) d x)

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

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

簡素化

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

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