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人気のある微分積分 >

integral of (sin^3(x))/(1+cos(x))

解

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

解

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

解答ステップ

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

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{x}{2} )} - (- \frac{1}{2 ( 1 + tan ^{2} ( \frac{x}{2} ) ) ^{2}}))

簡素化

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

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

定数を解答に追加する

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

グラフ

人気の例

integral of e^{-0.05t}(100t^4+10t^3)

\int e^{- 0.05 t} (100 t^{4} + 10 t^{3}) d t

integral of (ln(ln(y)))/y

\int \frac{ln ( ln ( y ) )}{y} d y

integral of e^{(x^2)^{22}}x^2)^2

\int e^{(x^{2})^{22}} d (x^{2})^{2}

integral of arctan(y/x)

\int arctan (\frac{y}{x}) d x

integral of (ln(x^2))/(x^4)

\int \frac{ln ( x ^{2} )}{x ^{4}} d x