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人気のある微分積分 >

integral of (sin^7(4x))/(cos^4(4x))

解

\int \frac{sin ^{7} ( 4 x )}{cos ^{4} ( 4 x )} d x

解

\frac{1}{12} (sec^{3} (4 x) - 9 sec (4 x) - 9 cos (4 x) + cos^{3} (4 x)) + C

解答ステップ

\int \frac{sin ^{7} ( 4 x )}{cos ^{4} ( 4 x )} d x

u置換積分法を適用する

= \int \frac{sin ^{7} ( u )}{4 cos ^{4} ( u )} 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 \frac{sin ^{7} ( u )}{cos ^{4} ( u )} d u

三角関数の公式を使用して書き換える

= \frac{1}{4} \cdot \int \frac{( 1 - cos ^{2} ( u ) ) ^{3} sin ( u )}{cos ^{4} ( u )} d u

u置換積分法を適用する

= \frac{1}{4} \cdot \int - \frac{( 1 - v ^{2} ) ^{3}}{v ^{4}} d v

拡張

- \frac{( 1 - v ^{2} ) ^{3}}{v ^{4}} : - \frac{1}{v ^{4}} + \frac{3}{v ^{2}} - 3 + v^{2}

= \frac{1}{4} \cdot \int - \frac{1}{v ^{4}} + \frac{3}{v ^{2}} - 3 + v^{2} d v

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{4} (- \int \frac{1}{v ^{4}} d v + \int \frac{3}{v ^{2}} d v - \int 3 d v + \int v^{2} d v)

\int \frac{1}{v ^{4}} d v = - \frac{1}{3 v ^{3}}

\int \frac{3}{v ^{2}} d v = - \frac{3}{v}

\int 3 d v = 3 v

\int v^{2} d v = \frac{v ^{3}}{3}

= \frac{1}{4} (- (- \frac{1}{3 v ^{3}}) - \frac{3}{v} - 3 v + \frac{v ^{3}}{3})

代用を戻す

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

簡素化

\frac{1}{4} (- (- \frac{1}{3 cos ^{3} ( 4 x )}) - \frac{3}{cos ( 4 x )} - 3 cos (4 x) + \frac{cos ^{3} ( 4 x )}{3}) : \frac{1}{12} (sec^{3} (4 x) - 9 sec (4 x) - 9 cos (4 x) + cos^{3} (4 x))

= \frac{1}{12} (sec^{3} (4 x) - 9 sec (4 x) - 9 cos (4 x) + cos^{3} (4 x))

定数を解答に追加する

= \frac{1}{12} (sec^{3} (4 x) - 9 sec (4 x) - 9 cos (4 x) + cos^{3} (4 x)) + C

グラフ

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