Calculus Examples

Evaluate the Integral integral of (x^4)/(x-1) with respect to x
Step 1
Divide by .
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Step 1.1
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 1.2
Divide the highest order term in the dividend by the highest order term in divisor .
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Step 1.3
Multiply the new quotient term by the divisor.
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Step 1.4
The expression needs to be subtracted from the dividend, so change all the signs in
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Step 1.5
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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Step 1.6
Pull the next terms from the original dividend down into the current dividend.
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Step 1.7
Divide the highest order term in the dividend by the highest order term in divisor .
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Step 1.8
Multiply the new quotient term by the divisor.
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Step 1.9
The expression needs to be subtracted from the dividend, so change all the signs in
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Step 1.10
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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Step 1.11
Pull the next terms from the original dividend down into the current dividend.
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Step 1.12
Divide the highest order term in the dividend by the highest order term in divisor .
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Step 1.13
Multiply the new quotient term by the divisor.
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Step 1.14
The expression needs to be subtracted from the dividend, so change all the signs in
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Step 1.15
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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Step 1.16
Pull the next terms from the original dividend down into the current dividend.
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Step 1.17
Divide the highest order term in the dividend by the highest order term in divisor .
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Step 1.18
Multiply the new quotient term by the divisor.
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Step 1.19
The expression needs to be subtracted from the dividend, so change all the signs in
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Step 1.20
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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Step 1.21
The final answer is the quotient plus the remainder over the divisor.
Step 2
Split the single integral into multiple integrals.
Step 3
By the Power Rule, the integral of with respect to is .
Step 4
By the Power Rule, the integral of with respect to is .
Step 5
By the Power Rule, the integral of with respect to is .
Step 6
Apply the constant rule.
Step 7
Let . Then . Rewrite using and .
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Step 7.1
Let . Find .
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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 .