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人気のある微分積分 >

integral of (1-cos(x))^{3/2}

解

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

解

2 \cdot 2^{\frac{1}{2}} (- 2 cos (\frac{x}{2}) + \frac{2}{3} cos^{3} (\frac{x}{2})) + C

解答ステップ

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

u置換積分法を適用する

= \int \frac{4 \cdot 2 ^{\frac{1}{2}} u ^{3}}{( 1 + u ^{2} ) ^{\frac{5}{2}}} d u

定数を除く:

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

= 4 \cdot 2^{\frac{1}{2}} \cdot \int \frac{u ^{3}}{( 1 + u ^{2} ) ^{\frac{5}{2}}} d u

u置換積分法を適用する

= 4 \cdot 2^{\frac{1}{2}} \cdot \int \frac{v - 1}{2 v ^{\frac{5}{2}}} d v

定数を除く:

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

= 4 \cdot 2^{\frac{1}{2}} \cdot \frac{1}{2} \cdot \int \frac{v - 1}{v ^{\frac{5}{2}}} d v

拡張

\frac{v - 1}{v ^{\frac{5}{2}}} : \frac{1}{v ^{\frac{3}{2}}} - \frac{1}{v ^{\frac{5}{2}}}

= 4 \cdot 2^{\frac{1}{2}} \cdot \frac{1}{2} \cdot \int \frac{1}{v ^{\frac{3}{2}}} - \frac{1}{v ^{\frac{5}{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

= 4 \cdot 2^{\frac{1}{2}} \cdot \frac{1}{2} (\int \frac{1}{v ^{\frac{3}{2}}} d v - \int \frac{1}{v ^{\frac{5}{2}}} d v)

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

\int \frac{1}{v ^{\frac{5}{2}}} d v = - \frac{2}{3 v ^{\frac{3}{2}}}

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

代用を戻す

= 4 \cdot 2^{\frac{1}{2}} \cdot \frac{1}{2} - \frac{2}{( 1 + tan ^{2} ( \frac{x}{2} ) ) ^{\frac{1}{2}}} - - \frac{2}{3 ( 1 + tan ^{2} ( \frac{x}{2} ) ) ^{\frac{3}{2}}}

簡素化

4 \cdot 2^{\frac{1}{2}} \cdot \frac{1}{2} - \frac{2}{( 1 + tan ^{2} ( \frac{x}{2} ) ) ^{\frac{1}{2}}} - - \frac{2}{3 ( 1 + tan ^{2} ( \frac{x}{2} ) ) ^{\frac{3}{2}}} : 2 \cdot 2^{\frac{1}{2}} (- 2 cos (\frac{x}{2}) + \frac{2}{3} cos^{3} (\frac{x}{2}))

= 2 \cdot 2^{\frac{1}{2}} (- 2 cos (\frac{x}{2}) + \frac{2}{3} cos^{3} (\frac{x}{2}))

定数を解答に追加する

= 2 \cdot 2^{\frac{1}{2}} (- 2 cos (\frac{x}{2}) + \frac{2}{3} cos^{3} (\frac{x}{2})) + C

グラフ

人気の例

integral of 5/(sqrt(t))

\int \frac{5}{t} d t

integral of 1-ln(y)

\int 1 - ln (y) d y

integral of x/(5-5x^3)

\int \frac{x}{5 - 5 x ^{3}} d x

integral of e^{-0.06t}

\int e^{- 0.06 t} d t

integral of 5/((2x-1)^2)

\int \frac{5}{( 2 x - 1 ) ^{2}} d x