Calculus Examples

Evaluate the Integral integral of 1/(x^2-1) with respect to x
Step 1
Write the fraction using partial fraction decomposition.
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Step 1.1
Decompose the fraction and multiply through by the common denominator.
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Step 1.1.1
Factor the fraction.
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Step 1.1.1.1
Rewrite as .
Step 1.1.1.2
Since both terms are perfect squares, factor using the difference of squares formula, where and .
Step 1.1.2
For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place .
Step 1.1.3
For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place .
Step 1.1.4
Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is .
Step 1.1.5
Cancel the common factor of .
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Step 1.1.5.1
Cancel the common factor.
Step 1.1.5.2
Rewrite the expression.
Step 1.1.6
Cancel the common factor of .
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Step 1.1.6.1
Cancel the common factor.
Step 1.1.6.2
Rewrite the expression.
Step 1.1.7
Simplify each term.
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Step 1.1.7.1
Cancel the common factor of .
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Step 1.1.7.1.1
Cancel the common factor.
Step 1.1.7.1.2
Divide by .
Step 1.1.7.2
Apply the distributive property.
Step 1.1.7.3
Move to the left of .
Step 1.1.7.4
Rewrite as .
Step 1.1.7.5
Cancel the common factor of .
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Step 1.1.7.5.1
Cancel the common factor.
Step 1.1.7.5.2
Divide by .
Step 1.1.7.6
Apply the distributive property.
Step 1.1.7.7
Multiply by .
Step 1.1.8
Move .
Step 1.2
Create equations for the partial fraction variables and use them to set up a system of equations.
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Step 1.2.1
Create an equation for the partial fraction variables by equating the coefficients of from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.
Step 1.2.2
Create an equation for the partial fraction variables by equating the coefficients of the terms not containing . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.
Step 1.2.3
Set up the system of equations to find the coefficients of the partial fractions.
Step 1.3
Solve the system of equations.
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Step 1.3.1
Solve for in .
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Step 1.3.1.1
Rewrite the equation as .
Step 1.3.1.2
Subtract from both sides of the equation.
Step 1.3.2
Replace all occurrences of with in each equation.
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Step 1.3.2.1
Replace all occurrences of in with .
Step 1.3.2.2
Simplify the right side.
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Step 1.3.2.2.1
Simplify .
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Step 1.3.2.2.1.1
Multiply .
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Step 1.3.2.2.1.1.1
Multiply by .
Step 1.3.2.2.1.1.2
Multiply by .
Step 1.3.2.2.1.2
Add and .
Step 1.3.3
Solve for in .
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Step 1.3.3.1
Rewrite the equation as .
Step 1.3.3.2
Divide each term in by and simplify.
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Step 1.3.3.2.1
Divide each term in by .
Step 1.3.3.2.2
Simplify the left side.
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Step 1.3.3.2.2.1
Cancel the common factor of .
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Step 1.3.3.2.2.1.1
Cancel the common factor.
Step 1.3.3.2.2.1.2
Divide by .
Step 1.3.4
Replace all occurrences of with in each equation.
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Step 1.3.4.1
Replace all occurrences of in with .
Step 1.3.4.2
Simplify the right side.
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Step 1.3.4.2.1
Multiply by .
Step 1.3.5
List all of the solutions.
Step 1.4
Replace each of the partial fraction coefficients in with the values found for and .
Step 1.5
Simplify.
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Step 1.5.1
Multiply the numerator by the reciprocal of the denominator.
Step 1.5.2
Multiply by .
Step 1.5.3
Move to the left of .
Step 1.5.4
Multiply the numerator by the reciprocal of the denominator.
Step 1.5.5
Multiply by .
Step 2
Split the single integral into multiple integrals.
Step 3
Since is constant with respect to , move out of the integral.
Step 4
Since is constant with respect to , move out of the integral.
Step 5
Let . Then . Rewrite using and .
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Step 5.1
Let . Find .
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Step 5.1.1
Differentiate .
Step 5.1.2
By the Sum Rule, the derivative of with respect to is .
Step 5.1.3
Differentiate using the Power Rule which states that is where .
Step 5.1.4
Since is constant with respect to , the derivative of with respect to is .
Step 5.1.5
Add and .
Step 5.2
Rewrite the problem using and .
Step 6
The integral of with respect to is .
Step 7
Since is constant with respect to , move out of the integral.
Step 8
Let . Then . Rewrite using and .
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Step 8.1
Let . Find .
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Step 8.1.1
Differentiate .
Step 8.1.2
By the Sum Rule, the derivative of with respect to is .
Step 8.1.3
Differentiate using the Power Rule which states that is where .
Step 8.1.4
Since is constant with respect to , the derivative of with respect to is .
Step 8.1.5
Add and .
Step 8.2
Rewrite the problem using and .
Step 9
The integral of with respect to is .
Step 10
Simplify.
Step 11
Substitute back in for each integration substitution variable.
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Step 11.1
Replace all occurrences of with .
Step 11.2
Replace all occurrences of with .