常用三角函数公式

常用三角函数公式

积分

符号情形

$$\int x\sin \left(x\right)dx=\sin (x)-x\cos (x)$$

$$\int x^2\sin \left(x\right)dx=2x\sin (x)-\left(x^2-2\right)\cos (x)$$

$$\int x^3\sin \left(x\right)dx=3\left(x^2-2\right)\sin (x)-x\left(x^2-6\right)\cos (x)$$

$$\int x\cos \left(x\right)dx=x\sin (x)+\cos (x)$$

$$\int x^2\cos \left(x\right)dx=\left(x^2-2\right)\sin (x)+2x\cos (x)$$

$$\int x^3\cos \left(x\right)dx=x\left(x^2-6\right)\sin (x)+3\left(x^2-2\right)\cos (x)$$

$$\int {\sin }^2\left(x\right)dx=\frac{x}{2}-\frac{1}{4}\sin (2x)$$

$$\int x{\sin }^2\left(x\right)dx=\frac{x^2}{4}-\frac{1}{4}x\sin (2x)-\frac{1}{8}\cos (2x)$$

$$\int {\cos }^2\left(x\right)dx=\frac{x}{2}+\frac{1}{4}\sin (2x)$$

$$\int x{\cos }^2\left(x\right)dx=\frac{x^2}{4}+\frac{1}{4}x\sin (2x)+\frac{1}{8}\cos (2x)$$

数值情形

$${\int }_0^{\pi }\sin \left(x\right)dx=2$$

$${\int }_0^{\pi }x\sin \left(x\right)dx=\pi$$

$${\int }_0^{\pi }{x}^2\sin \left(x\right)dx={\pi }^2-4$$

$${\int }_0^{\pi }{x}^3\sin \left(x\right)dx={\pi }^3-6\pi$$

$${\int }_0^{\pi }x\cos \left(x\right)dx=-2$$

$${\int }_0^{\pi }{x}^2\cos \left(x\right)dx=-2\pi$$

$${\int }_0^{\pi }{x}^3\cos \left(x\right)dx=-3{\pi }^2+12$$

参考

• None