integral of (4xsin(y^2))/(y^3)

解

\int \frac{4 x sin ( y ^{2} )}{y ^{3}} d y

2 x (- \frac{sin ( y ^{2} )}{y ^{2}} + Ci (y^{2})) + C

解答ステップ

\int \frac{4 x sin ( y ^{2} )}{y ^{3}} d y

定数を除く:

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

= 4 x \cdot \int \frac{sin ( y ^{2} )}{y ^{3}} d y

指数の規則を適用する:

\frac{1}{a ^{b}} = a^{- b}

\frac{sin ( y ^{2} )}{y ^{3}} = (sin (y^{2})) y^{- 3}

= 4 x \cdot \int sin (y^{2}) y^{- 3} d y

u置換積分法を適用する

= 4 x \cdot \int \frac{sin ( u )}{2 u ^{2}} d u

定数を除く:

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

= 4 x \frac{1}{2} \cdot \int \frac{sin ( u )}{u ^{2}} d u

部分積分を適用する

= 4 x \frac{1}{2} (- \frac{sin ( u )}{u} - \int - \frac{cos ( u )}{u} d u)

\int - \frac{cos ( u )}{u} d u = - Ci (u)

= 4 x \frac{1}{2} (- \frac{sin ( u )}{u} - (- Ci (u)))

代用を戻す

u = y^{2}

= 4 x \frac{1}{2} (- \frac{sin ( y ^{2} )}{y ^{2}} - (- Ci (y^{2})))

簡素化

4 x \frac{1}{2} (- \frac{sin ( y ^{2} )}{y ^{2}} - (- Ci (y^{2}))) : 2 x (- \frac{sin ( y ^{2} )}{y ^{2}} + Ci (y^{2}))

= 2 x (- \frac{sin ( y ^{2} )}{y ^{2}} + Ci (y^{2}))

定数を解答に追加する

= 2 x (- \frac{sin ( y ^{2} )}{y ^{2}} + Ci (y^{2})) + C