(sinx)^2(cosx)^4=[1-(cosx)^2](cosx)^4=(cosx)^4-(cosx)^6
=[(1+cos2x)/2]^2-[(1+cos2x)/2]^3
=[(cos2x)^2+2cosx+1]/4-[(cos2x)^3+3(cos2x)^2+3cos2x+1]/8
={[(1+cos4x)/2]+2cosx+1]/4-{(cos6x+3cos2x)/4+3[(1+cos4x)/2]+3cos2x+1}/8
=(cos4x+4cos2x+3)/4-(cos6x+3cos2x+6+6cos4x+12cos2x+4)/32
=-cos6x/32+cos4x/32+17cos2x/32+7/16
所以原式=
sin6x/192-sin4x/128-17sin2x/64+7/16x
不定积分的公式
1、∫ a dx = ax + C,a和C都是常数
2、∫ x^a dx = [x^(a + 1)]/(a + 1) + C,其中a为常数且 a ≠ -1
3、∫ 1/x dx = ln|x| + C
4、∫ a^x dx = (1/lna)a^x + C,其中a > 0 且 a ≠ 1
5、∫ e^x dx = e^x + C
6、∫ cosx dx = sinx + C
7、∫ sinx dx = - cosx + C
8、∫ cotx dx = ln|sinx| + C = - ln|cscx| + C
如图:
原式 = (sinx)^2(cosx)^4 - ∫(cosx)^4 d(sinx)^2
= (sinx)^2(cosx)^4 - ∫[1 - (sinx)^2]d(sinx)^2
= (sinx)^2(cosx)^4 - ∫[1 + (sinx)^4 - 2(sinx)^2] d(sinx)^2
= (sinx)^2(cosx)^4 - (sinx)^2 - (1/3)(sinx)^6 + (sinx)^4 + c