# Evaluate integral from 0 to 1 of arcsin(x) with respect to x

Integrate by parts using the formula , where and .
Combine and .
Let . Then , so . Rewrite using and .
Let . Find .
Differentiate .
By the Sum Rule, the derivative of with respect to is .
Since is constant with respect to , the derivative of with respect to is .
Evaluate .
Since is constant with respect to , the derivative of with respect to is .
Differentiate using the Power Rule which states that is where .
Multiply by .
Subtract from .
Substitute the lower limit in for in .
Simplify.
Simplify each term.
Raising to any positive power yields .
Multiply by .
Substitute the upper limit in for in .
Simplify.
Simplify each term.
One to any power is one.
Multiply by .
Subtract from .
The values found for and will be used to evaluate the definite integral.
Rewrite the problem using , , and the new limits of integration.
Simplify.
Move the negative in front of the fraction.
Multiply and .
Move to the left of .
Since is constant with respect to , move out of the integral.
Simplify.
Multiply by .
Multiply by .
Since is constant with respect to , move out of the integral.
Apply basic rules of exponents.
Rewrite as .
Move out of the denominator by raising it to the power.
Multiply the exponents in .
Apply the power rule and multiply exponents, .
Combine and .
Move the negative in front of the fraction.
By the Power Rule, the integral of with respect to is .
Combine and .
Substitute and simplify.
Evaluate at and at .
Evaluate at and at .
Simplify.
Multiply by .
Multiply by .
Multiply by .
Rewrite as .
Apply the power rule and multiply exponents, .
Cancel the common factor of .
Cancel the common factor.
Rewrite the expression.
Evaluate the exponent.
Multiply by .
One to any power is one.
Multiply by .
Subtract from .
Cancel the common factor of and .
Factor out of .
Cancel the common factors.
Factor out of .
Cancel the common factor.
Rewrite the expression.
Divide by .
The exact value of is .
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Evaluate integral from 0 to 1 of arcsin(x) with respect to x