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人気のある微分積分 >

integral of sec^7(3x)

解

\int sec^{7} (3 x) d x

解

\frac{1}{144} (8 sec^{6} (3 x) sin (3 x) + 5 (2 sec^{4} (3 x) sin (3 x) + 3 (sec (3 x) tan (3 x) + ln ∣ tan (3 x) + sec (3 x) ∣))) + C

解答ステップ

\int sec^{7} (3 x) d x

u置換積分法を適用する

= \int sec^{7} (u) \frac{1}{3} d u

定数を除く:

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

= \frac{1}{3} \cdot \int sec^{7} (u) d u

積分換算を適用する

= \frac{1}{3} (\frac{sec ^{6} ( u ) sin ( u )}{6} + \frac{5}{6} \cdot \int sec^{5} (u) d u)

\int sec^{5} (u) d u = \frac{sec ^{4} ( u ) sin ( u )}{4} + \frac{3}{4} (\frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)

= \frac{1}{3} (\frac{sec ^{6} ( u ) sin ( u )}{6} + \frac{5}{6} (\frac{sec ^{4} ( u ) sin ( u )}{4} + \frac{3}{4} (\frac{1}{2} sec (u) tan (u) + \frac{1}{2} ln ∣ tan (u) + sec (u) ∣)))

代用を戻す

u = 3 x

= \frac{1}{3} (\frac{sec ^{6} ( 3 x ) sin ( 3 x )}{6} + \frac{5}{6} (\frac{sec ^{4} ( 3 x ) sin ( 3 x )}{4} + \frac{3}{4} (\frac{1}{2} sec (3 x) tan (3 x) + \frac{1}{2} ln ∣ tan (3 x) + sec (3 x) ∣)))

簡素化

\frac{1}{3} (\frac{sec ^{6} ( 3 x ) sin ( 3 x )}{6} + \frac{5}{6} (\frac{sec ^{4} ( 3 x ) sin ( 3 x )}{4} + \frac{3}{4} (\frac{1}{2} sec (3 x) tan (3 x) + \frac{1}{2} ln ∣ tan (3 x) + sec (3 x) ∣))) : \frac{1}{144} (8 sec^{6} (3 x) sin (3 x) + 5 (2 sec^{4} (3 x) sin (3 x) + 3 (sec (3 x) tan (3 x) + ln ∣ tan (3 x) + sec (3 x) ∣)))

= \frac{1}{144} (8 sec^{6} (3 x) sin (3 x) + 5 (2 sec^{4} (3 x) sin (3 x) + 3 (sec (3 x) tan (3 x) + ln ∣ tan (3 x) + sec (3 x) ∣)))

定数を解答に追加する

= \frac{1}{144} (8 sec^{6} (3 x) sin (3 x) + 5 (2 sec^{4} (3 x) sin (3 x) + 3 (sec (3 x) tan (3 x) + ln ∣ tan (3 x) + sec (3 x) ∣))) + C

グラフ

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