What is the antiderivative of (4)/sq.rt.(9 - x^2) dx?

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What is the antiderivative of (4)/sq. rt. (9 - x^2) dx?

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Answer 1 · 2018-02-01T18:59:34

# 4arc sin(x/3)#.

Let, #I=int4/sqrt(9-x^2)dx#.

Using the subst. #x=3sintheta," so that, "dx=3costhetad theta#.

#:. I=int4/sqrt(9-9sin^2theta)3costhetad theta#,

#=4int(3costheta)/(3costheta)d theta#,

#=4intd theta#,

#=4theta#.

Since, #x=3sintheta, theta=arc sin(x/3)#.

#rArr I=4arc sin(x/3)#.