∫xcos²xdx的积分 求详细步骤 x²cosx的对x积分怎么算？求详细步骤
\u222bxcos^2xdx \u7684\u79ef\u5206(x*sin(2*x))/4 - sin(x)^2/4 + x^2/4
\u222bx²cosxdx=(x²-2)sinx+2xcosx+C\u3002\uff08C\u4e3a\u79ef\u5206\u5e38\u6570\uff09
\u89e3\u7b54\u8fc7\u7a0b\u5982\u4e0b\uff1a
\u222bx²cosxdx
=\u222bx²d(sinx)
=x²sinx-\u222bsinxd(x²)
=x²sinx-\u222b2xsinxdx
=x²sinx-2\u222bxd(-cosx)
=x²sinx+2\u222bxd(cosx)
=x²sinx+2xcosx-2\u222bcosxdx
=x²sinx+2xcosx-2sinx +C
=(x²-2)sinx+2xcosx+C
\u6269\u5c55\u8d44\u6599\uff1a
\u5206\u90e8\u79ef\u5206\u6cd5\uff1a\u8bbe\u51fd\u6570\u548cu\uff0cv\u5177\u6709\u8fde\u7eed\u5bfc\u6570\uff0c\u5219d(uv)=udv+vdu\u3002\u79fb\u9879\u5f97\u5230udv=d(uv)-vdu\u3002
\u4e24\u8fb9\u79ef\u5206\uff0c\u5f97\u5206\u90e8\u79ef\u5206\u516c\u5f0f\uff1a\u222budv=uv-\u222bvdu\u3002
\u5e38\u7528\u79ef\u5206\u516c\u5f0f\uff1a
1\uff09\u222b0dx=c
2\uff09\u222bx^udx=(x^(u+1))/(u+1)+c
3\uff09\u222b1/xdx=ln|x|+c
4\uff09\u222ba^xdx=(a^x)/lna+c
5\uff09\u222be^xdx=e^x+c
6\uff09\u222bsinxdx=-cosx+c
7\uff09\u222bcosxdx=sinx+c
8\uff09\u222b1/(cosx)^2dx=tanx+c
9\uff09\u222b1/(sinx)^2dx=-cotx+c
10\uff09\u222b1/\u221a\uff081-x^2) dx=arcsinx+c
cos²x=1+cos2x
∫xcos²xdx
=∫x(1+cos2x)dx
=∫xdx+∫xcos2xdx
=x²/2+x*sin2x/2-∫(sin2x)/2dx
=x²/2+x*sin2x/2+(cos2x)/4+c
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
cos²x=1+cos2x
∫xcos²xdx
=∫x(1+cos2x)dx
=∫xdx+∫xcos2xdx
=x²/2+x*sin2x/2-∫(sin2x)/2dx
=x²/2+x*sin2x/2+(cos2x)/4+c
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谢谢采纳
∫xcos²xdx=∫x(1+cos2x)/2dx
=∫x/2dx+∫(cos2x)/2dx
=x²/4+∫(cos2x)/4d2x
=x²/4+sin2x/4+c
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