Calculus Examples

Evaluate the Integral integral of 1/(x(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
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 is 2nd order, terms are required in the numerator. The number of terms required in the numerator is always equal to the order of the factor in the denominator.
Step 1.1.2
Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is .
Step 1.1.3
Cancel the common factor of .
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Step 1.1.3.1
Cancel the common factor.
Step 1.1.3.2
Rewrite the expression.
Step 1.1.4
Cancel the common factor of .
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Step 1.1.4.1
Cancel the common factor.
Step 1.1.4.2
Rewrite the expression.
Step 1.1.5
Simplify each term.
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Step 1.1.5.1
Cancel the common factor of .
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Step 1.1.5.1.1
Cancel the common factor.
Step 1.1.5.1.2
Divide by .
Step 1.1.5.2
Apply the distributive property.
Step 1.1.5.3
Multiply by .
Step 1.1.5.4
Cancel the common factor of .
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Step 1.1.5.4.1
Cancel the common factor.
Step 1.1.5.4.2
Divide by .
Step 1.1.5.5
Apply the distributive property.
Step 1.1.5.6
Multiply by by adding the exponents.
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Step 1.1.5.6.1
Move .
Step 1.1.5.6.2
Multiply by .
Step 1.1.6
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 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 1.2.3
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 1.2.4
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
Rewrite the equation as .
Step 1.3.2
Rewrite the equation as .
Step 1.3.3
Replace all occurrences of with in each equation.
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Step 1.3.3.1
Replace all occurrences of in with .
Step 1.3.3.2
Simplify the right side.
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Step 1.3.3.2.1
Remove parentheses.
Step 1.3.4
Solve for in .
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Step 1.3.4.1
Rewrite the equation as .
Step 1.3.4.2
Subtract from both sides of the equation.
Step 1.3.5
Solve the system of equations.
Step 1.3.6
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
Remove parentheses.
Step 1.5.2
Simplify the numerator.
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Step 1.5.2.1
Rewrite as .
Step 1.5.2.2
Add and .
Step 1.5.3
Move the negative in front of the fraction.
Step 2
Split the single integral into multiple integrals.
Step 3
The integral of with respect to is .
Step 4
Since is constant with respect to , move out of the integral.
Step 5
Let . Then , so . 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
Simplify.
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Step 6.1
Multiply by .
Step 6.2
Move to the left of .
Step 7
Since is constant with respect to , move out of the integral.
Step 8
The integral of with respect to is .
Step 9
Simplify.
Step 10
Replace all occurrences of with .