Calculus Examples

Evaluate the Integral integral of 1/(4-x^2) 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 using the commutative property of multiplication.
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
Move to the left of .
Step 1.1.8
Simplify the expression.
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Step 1.1.8.1
Move .
Step 1.1.8.2
Reorder and .
Step 1.1.8.3
Move .
Step 1.1.8.4
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
Rewrite as .
Step 1.3.1.3
Add to 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
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 left side.
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Step 1.3.4.2.1
Remove parentheses.
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
Multiply the numerator by the reciprocal of the denominator.
Step 1.5.4
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
Let . Then . Rewrite using and .
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Step 4.1
Let . Find .
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Step 4.1.1
Differentiate .
Step 4.1.2
By the Sum Rule, the derivative of with respect to is .
Step 4.1.3
Since is constant with respect to , the derivative of with respect to is .
Step 4.1.4
Differentiate using the Power Rule which states that is where .
Step 4.1.5
Add and .
Step 4.2
Rewrite the problem using and .
Step 5
The integral of with respect to is .
Step 6
Since is constant with respect to , move out of the integral.
Step 7
Let . Then , so . Rewrite using and .
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Step 7.1
Let . Find .
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Step 7.1.1
Rewrite.
Step 7.1.2
Divide by .
Step 7.2
Rewrite the problem using and .
Step 8
Move the negative in front of the fraction.
Step 9
Since is constant with respect to , move out of the integral.
Step 10
The integral of with respect to is .
Step 11
Simplify.
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Step 11.1
Simplify.
Step 11.2
Combine and .
Step 12
Substitute back in for each integration substitution variable.
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Step 12.1
Replace all occurrences of with .
Step 12.2
Replace all occurrences of with .
Step 13
Reorder terms.