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受欢迎的微积分 >

integral of tsin(t^2)cos(t^2)

解答

\int t sin (t^{2}) cos (t^{2}) d t

解答

- \frac{1}{8} cos (2 t^{2}) + C

求解步骤

\int t sin (t^{2}) cos (t^{2}) d t

使用三角恒等式改写

= \int \frac{1}{2} t sin (2 t^{2}) d t

提出常数:

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

= \frac{1}{2} \cdot \int t sin (2 t^{2}) d t

使用换元积分法

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

提出常数:

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

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

使用积分常见定则:

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

= \frac{1}{2} \cdot \frac{1}{4} (- cos (u))

u = 2 t^{2}

代回

= \frac{1}{2} \cdot \frac{1}{4} (- cos (2 t^{2}))

化简

\frac{1}{2} \cdot \frac{1}{4} (- cos (2 t^{2})) : - \frac{1}{8} cos (2 t^{2})

= - \frac{1}{8} cos (2 t^{2})

解答补常数

= - \frac{1}{8} cos (2 t^{2}) + C

作图

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