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^3(θ))/(1+cos(θ))

解

\int \frac{sin ^{3} ( θ )}{1 + cos ( θ )} d θ

解

4 (- cos^{2} (\frac{θ}{2}) + \frac{1}{2} cos^{4} (\frac{θ}{2})) + C

解答ステップ

\int \frac{sin ^{3} ( θ )}{1 + cos ( θ )} d θ

u置換積分法を適用する

= \int \frac{8 u ^{3}}{( 1 + u ^{2} ) ^{3}} d u

定数を除く:

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

= 8 \cdot \int \frac{u ^{3}}{( 1 + u ^{2} ) ^{3}} d u

u置換積分法を適用する

= 8 \cdot \int \frac{v - 1}{2 v ^{3}} d v

定数を除く:

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

= 8 \cdot \frac{1}{2} \cdot \int \frac{v - 1}{v ^{3}} d v

拡張

\frac{v - 1}{v ^{3}} : \frac{1}{v ^{2}} - \frac{1}{v ^{3}}

= 8 \cdot \frac{1}{2} \cdot \int \frac{1}{v ^{2}} - \frac{1}{v ^{3}} 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

= 8 \cdot \frac{1}{2} (\int \frac{1}{v ^{2}} d v - \int \frac{1}{v ^{3}} d v)

\int \frac{1}{v ^{2}} d v = - \frac{1}{v}

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

= 8 \cdot \frac{1}{2} (- \frac{1}{v} - (- \frac{1}{2 v ^{2}}))

代用を戻す

= 8 \cdot \frac{1}{2} (- \frac{1}{1 + tan ^{2} ( \frac{θ}{2} )} - (- \frac{1}{2 ( 1 + tan ^{2} ( \frac{θ}{2} ) ) ^{2}}))

簡素化

8 \cdot \frac{1}{2} (- \frac{1}{1 + tan ^{2} ( \frac{θ}{2} )} - (- \frac{1}{2 ( 1 + tan ^{2} ( \frac{θ}{2} ) ) ^{2}})) : 4 (- cos^{2} (\frac{θ}{2}) + \frac{1}{2} cos^{4} (\frac{θ}{2}))

= 4 (- cos^{2} (\frac{θ}{2}) + \frac{1}{2} cos^{4} (\frac{θ}{2}))

定数を解答に追加する

= 4 (- cos^{2} (\frac{θ}{2}) + \frac{1}{2} cos^{4} (\frac{θ}{2})) + C

グラフ

人気の例

integral of (3x-2)(x+1)^3

\int (3 x - 2) (x + 1)^{3} d x

integral of (x/3)^2

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

integral of sqrt(4+2x)

\int 4 + 2 x d x

integral of x^{1000}ln(x)

\int x^{1000} ln (x) d x

integral of 5x^4-5/(x^2)

\int 5 x^{4} - \frac{5}{x ^{2}} d x