# Evaluate integral from 0 to 1 of arctan(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 .
Differentiate using the Power Rule which states that is where .
Since is constant with respect to , the derivative of with respect to is .
Substitute the lower limit in for in .
Simplify.
Raising to any positive power yields .
Substitute the upper limit in for in .
Simplify.
One to any power is one.
The values found for and will be used to evaluate the definite integral.
Rewrite the problem using , , and the new limits of integration.
Simplify.
Multiply and .
Move to the left of .
Since is constant with respect to , move out of the integral.
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 .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Move to the left of .
Evaluate.
Simplify the numerator.
The exact value of is .
Cancel the common factor of .
Factor out of .
Cancel the common factor.
Rewrite the expression.
Use the quotient property of logarithms, .
The absolute value is the distance between a number and zero. The distance between and is .
The absolute value is the distance between a number and zero. The distance between and is .
Divide by .
To write as a fraction with a common denominator, multiply by .
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
Combine.
Multiply by .
Combine the numerators over the common denominator.
Rewrite in a factored form.
Multiply .
Multiply by .
Simplify by moving inside the logarithm.
Raise to the power of .
Multiply the numerator by the reciprocal of the denominator.
Multiply .
Multiply and .
Multiply by .
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Evaluate integral from 0 to 1 of arctan(x) with respect to x