Calculus Examples

Factor the numerator and denominator of .
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.
The integral of with respect to is .
Since is constant with respect to , move out of the integral.
Let . Then , so . Rewrite using and .
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Let . Find .
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Rewrite.
Divide by .
Rewrite the problem using and .
Combine fractions.
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Multiply and .
Move to the left of .
Since is constant with respect to , move out of the integral.
Combine fractions.
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Multiply and .
Multiply by .
The integral of with respect to is .
Since is constant with respect to , move out of the integral.
Since is constant with respect to , move out of the integral.
Let . Then . Rewrite using and .
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Let . Find .
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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 .
Add and .
Rewrite the problem using and .
The integral of with respect to is .
Simplify.
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Simplify.
Combine and .
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 .
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Combine.
Multiply by .
Combine the numerators over the common denominator.
Combine and .
Combine and .
Move to the left of .
Reduce the expression by cancelling the common factors.
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Cancel the common factor.
Divide by .
Replace all occurrences of with .
Replace all occurrences of with .
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