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Calculus Examples
Step 1
Integrate by parts using the formula , where and .
Step 2
Combine and .
Step 3
Step 3.1
Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of .
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Step 3.2
Divide the highest order term in the dividend by the highest order term in divisor .
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Step 3.3
Multiply the new quotient term by the divisor.
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Step 3.4
The expression needs to be subtracted from the dividend, so change all the signs in
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Step 3.5
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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Step 3.6
The final answer is the quotient plus the remainder over the divisor.
Step 4
Split the single integral into multiple integrals.
Step 5
Apply the constant rule.
Step 6
Since is constant with respect to , move out of the integral.
Step 7
Step 7.1
Let . Find .
Step 7.1.1
Differentiate .
Step 7.1.2
By the Sum Rule, the derivative of with respect to is .
Step 7.1.3
Differentiate using the Power Rule which states that is where .
Step 7.1.4
Since is constant with respect to , the derivative of with respect to is .
Step 7.1.5
Add and .
Step 7.2
Rewrite the problem using and .
Step 8
The integral of with respect to is .
Step 9
Simplify.
Step 10
Replace all occurrences of with .