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人気のある微分積分 >

integral of 3x^3cos(x^2)

解

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

解

\frac{3}{2} (x^{2} sin (x^{2}) + cos (x^{2})) + C

解答ステップ

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

定数を除く:

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

= 3 \cdot \int x^{3} cos (x^{2}) d x

u置換積分法を適用する

= 3 \cdot \int \frac{u cos ( u )}{2} d u

定数を除く:

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

= 3 \cdot \frac{1}{2} \cdot \int u cos (u) d u

部分積分を適用する

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

\int sin (u) d u = - cos (u)

= 3 \cdot \frac{1}{2} (u sin (u) - (- cos (u)))

代用を戻す

u = x^{2}

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

簡素化

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

= \frac{3}{2} (x^{2} sin (x^{2}) + cos (x^{2}))

定数を解答に追加する

= \frac{3}{2} (x^{2} sin (x^{2}) + cos (x^{2})) + C

グラフ

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