Calculus Examples

Evaluate the Integral integral of natural log of 1-x with respect to x
Step 1
Integrate by parts using the formula , where and .
Step 2
Combine and .
Step 3
Since is constant with respect to , move out of the integral.
Step 4
Simplify the expression.
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Step 4.1
Simplify.
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Step 4.1.1
Multiply by .
Step 4.1.2
Multiply by .
Step 4.2
Reorder and .
Step 5
Divide by .
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Step 5.1
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 5.2
Divide the highest order term in the dividend by the highest order term in divisor .
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Step 5.3
Multiply the new quotient term by the divisor.
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Step 5.4
The expression needs to be subtracted from the dividend, so change all the signs in
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Step 5.5
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
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Step 5.6
The final answer is the quotient plus the remainder over the divisor.
Step 6
Split the single integral into multiple integrals.
Step 7
Apply the constant rule.
Step 8
Let . Then , so . Rewrite using and .
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Step 8.1
Let . Find .
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Step 8.1.1
Rewrite.
Step 8.1.2
Divide by .
Step 8.2
Rewrite the problem using and .
Step 9
Move the negative in front of the fraction.
Step 10
Since is constant with respect to , move out of the integral.
Step 11
The integral of with respect to is .
Step 12
Simplify.
Step 13
Replace all occurrences of with .