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Calculus Examples
Step 1
Integrate by parts using the formula , where and .
Step 2
Step 2.1
Combine and .
Step 2.2
Combine and .
Step 3
Since is constant with respect to , move out of the integral.
Step 4
Step 4.1
Combine and .
Step 4.2
Move to the left of .
Step 5
Since is constant with respect to , move out of the integral.
Step 6
Step 6.1
Multiply by .
Step 6.2
Combine and .
Step 6.3
Move the negative in front of the fraction.
Step 7
Step 7.1
Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of .
+ | + | + | + |
Step 7.2
Divide the highest order term in the dividend by the highest order term in divisor .
+ | + | + | + |
Step 7.3
Multiply the new quotient term by the divisor.
+ | + | + | + | ||||||||
+ | + | + |
Step 7.4
The expression needs to be subtracted from the dividend, so change all the signs in
+ | + | + | + | ||||||||
- | - | - |
Step 7.5
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
+ | + | + | + | ||||||||
- | - | - | |||||||||
- |
Step 7.6
The final answer is the quotient plus the remainder over the divisor.
Step 8
Split the single integral into multiple integrals.
Step 9
Apply the constant rule.
Step 10
Since is constant with respect to , move out of the integral.
Step 11
Since is constant with respect to , move out of the integral.
Step 12
Reorder and .
Step 13
Step 13.1
Factor out of .
Step 13.2
Factor out of .
Step 13.3
Factor out of .
Step 14
Since is constant with respect to , move out of the integral.
Step 15
Step 15.1
Multiply by .
Step 15.2
Multiply by .
Step 16
Rewrite as .
Step 17
The integral of with respect to is .
Step 18
Step 18.1
Simplify.
Step 18.1.1
Multiply by the reciprocal of the fraction to divide by .
Step 18.1.2
Multiply by .
Step 18.1.3
Multiply by the reciprocal of the fraction to divide by .
Step 18.1.4
Move to the left of .
Step 18.2
Simplify.
Step 18.3
Simplify.
Step 18.3.1
To write as a fraction with a common denominator, multiply by .
Step 18.3.2
Combine and .
Step 18.3.3
Combine the numerators over the common denominator.
Step 18.3.4
Multiply by .
Step 18.3.5
Combine and .
Step 18.3.6
Multiply by .
Step 18.3.7
Cancel the common factor of and .
Step 18.3.7.1
Factor out of .
Step 18.3.7.2
Cancel the common factors.
Step 18.3.7.2.1
Factor out of .
Step 18.3.7.2.2
Cancel the common factor.
Step 18.3.7.2.3
Rewrite the expression.
Step 18.3.7.2.4
Divide by .
Step 18.4
Simplify.
Step 18.4.1
Apply the distributive property.
Step 18.4.2
Cancel the common factor of .
Step 18.4.2.1
Factor out of .
Step 18.4.2.2
Factor out of .
Step 18.4.2.3
Cancel the common factor.
Step 18.4.2.4
Rewrite the expression.
Step 18.4.3
Cancel the common factor of .
Step 18.4.3.1
Move the leading negative in into the numerator.
Step 18.4.3.2
Factor out of .
Step 18.4.3.3
Factor out of .
Step 18.4.3.4
Cancel the common factor.
Step 18.4.3.5
Rewrite the expression.
Step 18.4.4
Simplify each term.
Step 18.4.4.1
Move the negative in front of the fraction.
Step 18.4.4.2
Multiply .
Step 18.4.4.2.1
Multiply by .
Step 18.4.4.2.2
Multiply by .
Step 19
Reorder terms.