# Evaluate integral from 0 to 2 of 1/(1-x^2) with respect to x

Factor the numerator and denominator of .
Rewrite as .
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Write the fraction using partial fraction decomposition.
Simplify.
Split the single integral into multiple integrals.
Since is constant with respect to , move out of the integral.
Let . Then . 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 .
Differentiate using the Power Rule which states that is where .
Substitute the lower limit in for in .
Substitute the upper limit in for in .
The values found for and will be used to evaluate the definite integral.
Rewrite the problem using , , and the new limits of integration.
The integral of with respect to is .
Since is constant with respect to , move out of the integral.
Let . Then , so . Rewrite using and .
Let . Find .
Rewrite.
Divide by .
Substitute the lower limit in for in .
Subtract from .
Substitute the upper limit in for in .
Simplify.
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.
Move the negative in front of the fraction.
Since is constant with respect to , move out of the integral.
Simplify.
Combine and .
Move the negative in front of the fraction.
The integral of with respect to is .
Substitute and simplify.
Evaluate at and at .
Evaluate at and at .
Remove parentheses.
Evaluate.
Simplify each term.
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 .
Simplify by moving inside the logarithm.
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 .
The natural logarithm of is .
Multiply .
Multiply by .
Multiply by .