将多项式放在上侧，对其求导；

将 e a x e^{ax} eax或 sin ⁡ b x \sin bx sinbx或 cos ⁡ c x \cos cx coscx放在下侧，对其积分。

符号依次为：+，-，+，-，…

∫ 0 π x 2 cos ⁡ n x   d x = x 2 ⋅ 1 n sin ⁡ n x ∣ 0 π + 2 x ⋅ 1 n 2 cos ⁡ n x ∣ 0 π − 2 ⋅ 1 n 3 sin ⁡ n x ∣ 0 π = ( − 1 ) n 2 π n 2 . \int_{0}^{\pi}x^{2}\cos nx\ dx=x^{2}\cdot \frac{1}{n}\sin nx|_{0}^{\pi}+2x \cdot \frac{1}{n^{2}}\cos nx|_{0}^{\pi}-2 \cdot \frac{1}{n^{3}}\sin nx|_{0}^{\pi}=(-1)^{n}\frac{2 \pi}{n^{2}}. ∫0π​x2cosnx dx=x2⋅n1​sinnx∣0π​+2x⋅n21​cosnx∣0π​−2⋅n31​sinnx∣0π​=(−1)nn22π​.

∫ e a x sin ⁡ b x   d x = ∣ ( e a x ) ′ ( sin ⁡ b x ) ′ e a x sin ⁡ b x ∣ a 2   +   b 2 + C = a   e a x sin ⁡ b x   −   b   e a x cos ⁡ b x a 2   +   b 2 + C \int e^{ax} \sin bx\ dx=\frac{\begin{vmatrix} (e^{ax})' & (\sin bx)'\\ e^{ax}&\sin bx \end{vmatrix} }{a^{2}\ +\ b^{2}}+C =\frac{a\ e^{ax} \sin bx\ -\ b\ e^{ax}\cos bx}{a^{2}\ +\ b^{2} } +C ∫eaxsinbx dx=a2 + b2 ​(eax)′eax​(sinbx)′sinbx​ ​​+C=a2 + b2a eaxsinbx − b eaxcosbx​+C

∫ e a x cos ⁡ b x   d x = ∣ ( e a x ) ′ ( cos ⁡ b x ) ′ e a x cos ⁡ b x ∣ a 2   +   b 2 + C = a   e a x cos ⁡ b x   +   b   e a x sin ⁡ b x a 2   +   b 2 + C \int e^{ax} \cos bx\ dx=\frac{\begin{vmatrix} (e^{ax})' & (\cos bx)'\\ e^{ax}&\cos bx \end{vmatrix} }{a^{2}\ +\ b^{2}}+C =\frac{a\ e^{ax} \cos bx\ +\ b\ e^{ax}\sin bx}{a^{2}\ +\ b^{2} } +C ∫eaxcosbx dx=a2 + b2 ​(eax)′eax​(cosbx)′cosbx​ ​​+C=a2 + b2a eaxcosbx + b eaxsinbx​+C

∫ e − x sin ⁡ n x   d x = − e − x sin ⁡ n x − n   e − x cos ⁡ n x ( − 1 ) 2 + n 2 + C \int e^{-x} \sin nx\ dx=\frac{-e^{-x} \sin nx-n\ e^{-x}\cos nx}{(-1)^{2} +n^{2} } +C ∫e−xsinnx dx=(−1)2+n2−e−xsinnx−n e−xcosnx​+C