Question #dfe07

1 Answer

#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

#--------------------#
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)#

#--------------------#

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"#