sqrt(1-x^2)的积分

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sqrt(1-x^2)的积分

sqrt(1-x^2)的积分
sqrt(1-x^2)的积分

sqrt(1-x^2)的积分
令x=sint
∫sqrt(1-x^2)dx
=∫costdsint
=∫(cost)^2dt
=1/2∫(1+cos2t)dt
=t/2+sin2t/4
arcsinx/2+xsqrt(1-x^2)/2+C

-x(1-x^2)^(-1/2)

[(1-x^2)^1/2]'
=1/2(1-x^2)^(-1/2)*(1-x^2)'
=-x(1-x^2)^(-1/2)

设x=sin t(|x|<=1)(|t|<=pai/2),用[ 代替积分号吧,
[ (1-x^2)^1/2 *dx=[ (cost)^2 *dt=[ (1+cos 2t)*dt=t+1/2 *sin2t=t+sint *cost
代入x得,arcsin x+x*sqrt(1-x^2)

令x=sint
∫sqrt(1-x^2)dx
=∫costdsint
=∫(cost)^2dt
=1/2∫(1+cos2t)dt
=t/2+sin2t/4
arcsinx/2+xsqrt(1-x^2)/2+常数