Calculus Examples

Find the Antiderivative (x^2)/(x+1)
Step 1
Write as a function.
Step 2
The function can be found by finding the indefinite integral of the derivative .
Step 3
Set up the integral to solve.
Step 4
Divide by .
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Step 4.1
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 4.2
Divide the highest order term in the dividend by the highest order term in divisor .
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Step 4.3
Multiply the new quotient term by the divisor.
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Step 4.4
The expression needs to be subtracted from the dividend, so change all the signs in
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Step 4.5
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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Step 4.6
Pull the next terms from the original dividend down into the current dividend.
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Step 4.7
Divide the highest order term in the dividend by the highest order term in divisor .
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Step 4.8
Multiply the new quotient term by the divisor.
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Step 4.9
The expression needs to be subtracted from the dividend, so change all the signs in
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Step 4.10
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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Step 4.11
The final answer is the quotient plus the remainder over the divisor.
Step 5
Split the single integral into multiple integrals.
Step 6
By the Power Rule, the integral of with respect to is .
Step 7
Apply the constant rule.
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
Replace all occurrences of with .
Step 12
The answer is the antiderivative of the function .