integral of sin(ax)sin(bx)

解答

\int sin (a x) sin (b x) d x

\frac{1}{2} (- \frac{1}{a + b} sin (a x + b x) + \frac{1}{a - b} sin (a x - b x)) + C

求解步骤

\int sin (a x) sin (b x) d x

使用三角恒等式改写

= \int \frac{- cos ( a x + b x ) + cos ( a x - b x )}{2} 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 (a x + b x) + cos (a x - b 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 (a x + b x) d x + \int cos (a x - b x) d x)

\int cos (a x + b x) d x = \frac{1}{a + b} sin (a x + b x)

\int cos (a x - b x) d x = \frac{1}{a - b} sin (a x - b x)

= \frac{1}{2} (- \frac{1}{a + b} sin (a x + b x) + \frac{1}{a - b} sin (a x - b x))

解答补常数

= \frac{1}{2} (- \frac{1}{a + b} sin (a x + b x) + \frac{1}{a - b} sin (a x - b x)) + C

t an g e n t y = x^{4} + 5 e^{x}, (0, 5)

\frac{\partial}{\partial x} (2 x e^{8 x y})

d er i v a t i v e y = 7 e^{x} + \frac{4}{3 x}

x \to 0 lim (\frac{9 ^{x} - 1 0 ^{x}}{x})

\int 4 3 x d x