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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

提出常数:

\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

使用指数法则:

\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

使用换元积分法

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

提出常数:

\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

使用分布积分法

= 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)))

u = y^{2}

代回

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

化简

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}))

解答补常数

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

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