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

解答

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

\frac{1}{2} (x^{2} sin (x^{2}) + cos (x^{2})) + C

求解步骤

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

使用换元积分法

= \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

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

使用分布积分法

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

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

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

u = x^{2}

代回

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

化简

= \frac{1}{2} (x^{2} sin (x^{2}) + cos (x^{2}))

解答补常数

= \frac{1}{2} (x^{2} sin (x^{2}) + cos (x^{2})) + C

\frac{d}{d x} ((4 x^{3} - 7 x)^{7})

\int x 2 x - x^{2} d x

\int (\frac{8 + 5 x}{1 + x ^{2}}) d x

\frac{d}{d x} (4 x^{4} + x^{3} - \frac{9 x ^{2}}{2} + 8 x)

t a y l or sin (π x), \frac{1}{2}