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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

提出常数:

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

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

使用换元积分法

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

提出常数:

\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

使用分布积分法

= 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)))

u = x^{2}

代回

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

化简

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}))

解答补常数

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

作图

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