Question #dfe07

1 Answer
Jan 14, 2018

intarcsin(x)/sqrt(1-x^2)dx=arcsin^2(x)/2+"C"

Explanation:

Given: intarcsin(x)/sqrt(1-x^2)dx

Apply u-substitution

Let u=arcsin(x)

du=1/sqrt(1-x^2)dx

See Proof Below

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Let: y=arcsin(x)

Take the sin of both sides. So

sin(y)=x

Use implicit differentation to differentiate both sides

dy/dx*cos(y)=1

Divde by cos(y) on both sides

dy/dx=1/color(red)(cos(y)

Rewrite in terms of x

Since sin(y)=x/1

Then color(red)(cos(y)=sqrt(1^2-x^2)/1=sqrt(1-x^2)

So rewritng we get

dy/dx=1/color(red)(cos(y))=color(red)(1/sqrt(1-x^2)

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Thus, the integral now becomes:

intudu

Since intx^a=x^(a+1)/(a+1), we solve the integral to get:

=u^(1+1)/(1+1)=u^2/2

Reverse the subsitution:

=arcsin^2(x)/2+"C"