$$ 142. \,\int\!\! ^{\pi}_{-\pi} \cos nx dx = \int\!\! ^{\pi}_{\pi} \sin nx dx=0 $$

$$ 143. \,\int\!\! ^ {\pi} _{-\pi} \cos mx \sin nx dx =0 $$

$$ 144. \,\int\!\! ^{\pi} _{-\pi} \cos mx \cos nx dx =\left \{ \begin{array}{cc} 0, & m \neq n \\ \pi, & m=n \end{array} \right. $$

$$ 145. \,\int\!\! ^\pi _{-\pi} \sin mx \sin nx dx = \left \{ \begin{array}{cc} 0, & m \neq n \\ \pi, & m=n \end{array} \right. $$

$$ 146. \,\int\!\! ^\pi _0 \sin mx \sin nx dx= \int\!\! ^\pi _0 \cos mx \cos nx dx = \left \{ \begin{array}{cc} 0, & m \neq n \\ \pi/2, & m=n \end{array} \right. $$

$$ 147. I_n =\,\int\!\! ^{\frac{\pi}{2}} _0 \sin ^n x dx = \int\!\! ^{\frac{\pi}{2}} _0 \cos ^n x dx;\qquad I_n= \frac{n-1}{n} I _{n-2}$$

$$I_n= \frac{n-1}{n}.\frac{n-3}{n-2}.\dots \frac{4}{5} .\frac{2}{3}(n为大于1的正奇数),I_1=1 $$

$$ I_n= \frac{n-1}{n} . \frac{n-3}{n-2} . \dots \frac{3}{4} .\frac{1}{2}. \frac{\pi}{2} (n为正偶数),I_0= \frac{\pi}{2}$$