Question

Question #dfe07

Answer 1

`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"`