求不定积分y=根号(r^2-x^2),r是常数要详细过程∫y=根号(r^2-x^2)dx，r是常数

Question

求不定积分
y=根号(r^2-x^2),r是常数
要详细过程∫y=根号(r^2-x^2) dx，r是常数

Answer 1

令x=rsint
根号(r²-x²)=rcost,dx=rcostdt
∫根号(r²-x²)dx
=∫r²cos²tdt
=(r²/2)∫(1+cos2t)dt
=(r²/2)(t+(1/2)sin2t)+C
=(r²/2)(t+sintcost)
=(r²/2)(arcsin(x/r)+(x/r)(根号(r²-x²)/r))+C
=(r²/2)arcsin(x/r)+x(根号(r²-x²))/2+C