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人気のある微分積分 >

integral of sec^3(4x)

解

\int sec^{3} (4 x) d x

解

\frac{1}{4} (\frac{1}{2} sec (4 x) tan (4 x) + \frac{1}{2} ln ∣ tan (4 x) + sec (4 x) ∣) + C

解答ステップ

\int sec^{3} (4 x) d x

u置換積分法を適用する

= \int sec^{3} (u) \frac{1}{4} d u

定数を除く:

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

= \frac{1}{4} \cdot \int sec^{3} (u) d u

積分換算を適用する

= \frac{1}{4} (\frac{sec ^{2} ( u ) sin ( u )}{2} + \frac{1}{2} \cdot \int sec (u) d u)

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

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

代用を戻す

u = 4 x

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

簡素化

\frac{1}{4} (\frac{sec ^{2} ( 4 x ) sin ( 4 x )}{2} + \frac{1}{2} ln ∣ tan (4 x) + sec (4 x) ∣) : \frac{1}{4} (\frac{1}{2} sec (4 x) tan (4 x) + \frac{1}{2} ln ∣ tan (4 x) + sec (4 x) ∣)

= \frac{1}{4} (\frac{1}{2} sec (4 x) tan (4 x) + \frac{1}{2} ln ∣ tan (4 x) + sec (4 x) ∣)

定数を解答に追加する

= \frac{1}{4} (\frac{1}{2} sec (4 x) tan (4 x) + \frac{1}{2} ln ∣ tan (4 x) + sec (4 x) ∣) + C

グラフ

人気の例

derivative of 1 1/x

\frac{d}{d x} (1 \frac{1}{x})

derivative of (3x+1^6sqrt(2x-5))

\frac{d}{d x} ((3 x + 1)^{6} 2 x - 5)

integral of 2sqrt(x)

\int 2 x d x

d/(d{r)}({r}^3)

\frac{d}{d r} (r^{3})

f(x)=arctan(4x)

f (x) = arctan (4 x)