integral of sin(3x)sin(7x)

解答

\int sin (3 x) sin (7 x) d x

\frac{1}{2} (\frac{1}{4} sin (4 x) - \frac{1}{10} sin (10 x)) + C

求解步骤

\int sin (3 x) sin (7 x) d x

使用三角恒等式改写

= \int \frac{1}{2} (- cos (10 x) + cos (- 4 x)) d x

提出常数:

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

= \frac{1}{2} \cdot \int - cos (10 x) + cos (- 4 x) d x

化简

- cos (10 x) + cos (- 4 x) : cos (4 x) - cos (10 x)

= \frac{1}{2} \cdot \int cos (4 x) - cos (10 x) d x

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (\int cos (4 x) d x - \int cos (10 x) d x)

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

\int cos (10 x) d x = \frac{1}{10} sin (10 x)

= \frac{1}{2} (\frac{1}{4} sin (4 x) - \frac{1}{10} sin (10 x))

解答补常数

= \frac{1}{2} (\frac{1}{4} sin (4 x) - \frac{1}{10} sin (10 x)) + C

\frac{d}{d x} (- x^{\frac{2}{3}})

x \to 1 lim (\frac{x ^{3} + 1}{x + 1})

y^{^{'}} = \frac{x ^{2} + 3 x + 2}{y - 2}

\frac{d}{d x} (arctan^{2} (x^{3}))

\frac{\partial}{\partial y} (e^{y} \cdot sin (x))