ja

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

人気のある微分積分 >

integral of (sin^7(6x))/(cos^4(6x))

解

\int \frac{sin ^{7} ( 6 x )}{cos ^{4} ( 6 x )} d x

解

\frac{1}{18} (sec^{3} (6 x) - 9 sec (6 x) - 9 cos (6 x) + cos^{3} (6 x)) + C

解答ステップ

\int \frac{sin ^{7} ( 6 x )}{cos ^{4} ( 6 x )} d x

u置換積分法を適用する

= \int \frac{sin ^{7} ( u )}{6 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}{6} \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}{6} \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}{6} \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}{6} \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}{6} (- \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}{6} (- (- \frac{1}{3 v ^{3}}) - \frac{3}{v} - 3 v + \frac{v ^{3}}{3})

代用を戻す

= \frac{1}{6} (- (- \frac{1}{3 cos ^{3} ( 6 x )}) - \frac{3}{cos ( 6 x )} - 3 cos (6 x) + \frac{cos ^{3} ( 6 x )}{3})

簡素化

\frac{1}{6} (- (- \frac{1}{3 cos ^{3} ( 6 x )}) - \frac{3}{cos ( 6 x )} - 3 cos (6 x) + \frac{cos ^{3} ( 6 x )}{3}) : \frac{1}{18} (sec^{3} (6 x) - 9 sec (6 x) - 9 cos (6 x) + cos^{3} (6 x))

= \frac{1}{18} (sec^{3} (6 x) - 9 sec (6 x) - 9 cos (6 x) + cos^{3} (6 x))

定数を解答に追加する

= \frac{1}{18} (sec^{3} (6 x) - 9 sec (6 x) - 9 cos (6 x) + cos^{3} (6 x)) + C

グラフ

人気の例

limit as x approaches 2.9 of x^2+3x

x \to 2.9 lim (x^{2} + 3 x)

limit as x approaches-1 of x^2+1

x \to - 1 lim (x^{2} + 1)

integral of x*cos(2x)

\int x \cdot cos (2 x) d x

integral of (x+6)/((x+2)^2)

\int \frac{x + 6}{( x + 2 ) ^{2}} d x

derivative of insin(x)

\frac{d}{d x} (in sin (x))