① ∫(2x+4)/(x2 +2x+3) dx; ② ∫(x2)/(1+x2)arctanx dx; ③ 1/[(3√x)+1] dx 注:注:x后的2为平方,根号前的3为开立方;
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![① ∫(2x+4)/(x2 +2x+3) dx; ② ∫(x2)/(1+x2)arctanx dx; ③ 1/[(3√x)+1] dx 注:注:x后的2为平方,根号前的3为开立方;](/uploads/image/z/3694383-63-3.jpg?t=%E2%91%A0+%E2%88%AB%282x%2B4%29%EF%BC%8F%28x2+%2B2x%2B3%29+dx%EF%BC%9B+%E2%91%A1+%E2%88%AB%EF%BC%88x2%EF%BC%89%EF%BC%8F%281%2Bx2%29arctanx+dx%3B+%E2%91%A2+1%EF%BC%8F%5B%EF%BC%883%E2%88%9Ax%EF%BC%89%2B1%5D+dx+%E6%B3%A8%EF%BC%9A%E6%B3%A8%EF%BC%9Ax%E5%90%8E%E7%9A%842%E4%B8%BA%E5%B9%B3%E6%96%B9%2C%E6%A0%B9%E5%8F%B7%E5%89%8D%E7%9A%843%E4%B8%BA%E5%BC%80%E7%AB%8B%E6%96%B9%EF%BC%9B)
① ∫(2x+4)/(x2 +2x+3) dx; ② ∫(x2)/(1+x2)arctanx dx; ③ 1/[(3√x)+1] dx 注:注:x后的2为平方,根号前的3为开立方;
① ∫(2x+4)/(x2 +2x+3) dx; ② ∫(x2)/(1+x2)arctanx dx; ③ 1/[(3√x)+1] dx 注:
注:x后的2为平方,根号前的3为开立方;
① ∫(2x+4)/(x2 +2x+3) dx; ② ∫(x2)/(1+x2)arctanx dx; ③ 1/[(3√x)+1] dx 注:注:x后的2为平方,根号前的3为开立方;
1、原式=∫d(x^2+2x+3)/(x^2+2x+3)+2∫dx/(x^2+2x+3)
=ln|x^2+2x+3|+2∫dx/[(x+1)^2+2]
=ln|x^2+2x+3|+√2∫d[(x+1)/√2]/{[(x+1)/√2]^2+1}
=ln|x^2+2x+3|+√2arctan(x+1)/√2+C.
2、原式=∫(1+x^2-1)arctanxdx/(1+x^2)
=∫arctanxdx-∫arctanxdx/(1+x^2)
=x*arctanx-∫xdx/(1+x^2)-∫arctanxd(arctanx)
=x*arctanx-(1/2)∫d(1+x^2)/(1+x^2)-(1/2)(arctanx)^2
=xarctanx-(1/2)ln(1+x^2)-(1/2)(arctanx)^2+C.
3、设x^(1/3)=t,
x=t^3,
dx=3t^2dt,
原式=∫3t^2dt/(t+1)
=3∫(t+1)dt-6∫dt+3∫d(t+1)/(t+1)
=3(t^2/2+t)-6t+3ln|1+t|+C
=3[x^(2/3)/2+x^(1/3)]-6x^(1/3)+3ln|1+x^(1/3)|+C
=(3/2)x^(2/3) -3x^(1/3)+3ln|1+x^(1/3)|+C.