#int sqrt(9-x^2)d x=?#
#"Substitute "x=3*sin theta#
#d x=3*cos theta *d theta" ; "theta =arc sin(x/3)#
#x^2=9*sin^2 theta#
#int sqrt(9-x^2)d x=int sqrt(9-9*sin^2 theta)* 3*cos theta * d theta#
#int sqrt(9-x^2)d x=int sqrt(9(1-sin^2 theta))*3*cos theta*d theta#
#"So ;" 1-sin^2 theta =cos ^2theta#
#int sqrt(9-x^2)d x=int 3 sqrt(cos^2 theta)*3*cos theta*d theta#
#int sqrt(9-x^2)d x=9 int cos^2 theta *d theta#
#"Now use the reduction formula..."#
#int cos^n theta d theta=(n-1)/n int cos^(n-2) theta*d theta+(cos^(n-1)theta*sin theta)/n#
#"use " n=2#
#int cos^2 theta d theta=(2-1)/2 int cos^(2-2) theta*d theta +(cos^(2-1) theta*sin theta)/2#
#int cos^2 theta* d theta=1/2 int d theta+1/2(cos theta*sin theta) #
#int d theta =theta" ; "theta=arc sin (x/3)#
#"So ;"x=3*sin theta;" ; "sin theta=x/3#
#cos theta=sqrt(1-sin^2 theta)#
#cos theta=sqrt(1-x^2/9)#
#int sqrt(9-x^2)d x=9[1/2*arc sin (x/3)+1/2(sqrt(1-x^2/9 )*x/3)]#
#int sqrt(9-x^2)d x=(9 *arc sin (x/3))/2+1/2(sqrt(1-x^2/9)*3x)#
#int sqrt(9-x^2)d x=(9 arc sin (x/3))/2+1/2(sqrt((9-x^2)/9)*3x)#
#int sqrt(9-x^2)d x=(9 arc sin (x/3))/2+1/2(sqrt(9-x^2)*x)#
#"The problem has solved..."#
#int sqrt(9-x^2)d x=(9 arc sin (x/3)+x*sqrt(9-x^2))/2+C#
