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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
Multiply by .
Step 4.2
Combine and .
Step 4.3
Move the negative in front of the fraction.
Step 4.4
Multiply by .
Step 4.5
Multiply by .
Step 5
Integrate by parts using the formula , where and .
Step 6
Step 6.1
Combine and .
Step 6.2
Combine and .
Step 6.3
Combine and .
Step 7
Since is constant with respect to , move out of the integral.
Step 8
Step 8.1
Let . Find .
Step 8.1.1
Differentiate .
Step 8.1.2
Since is constant with respect to , 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
Multiply by .
Step 8.2
Rewrite the problem using and .
Step 9
Combine and .
Step 10
Since is constant with respect to , move out of the integral.
Step 11
Step 11.1
Multiply by .
Step 11.2
Raise to the power of .
Step 11.3
Raise to the power of .
Step 11.4
Use the power rule to combine exponents.
Step 11.5
Add and .
Step 12
The integral of with respect to is .
Step 13
Step 13.1
Rewrite as .
Step 13.2
Simplify.
Step 13.2.1
To write as a fraction with a common denominator, multiply by .
Step 13.2.2
Combine and .
Step 13.2.3
Combine the numerators over the common denominator.
Step 13.2.4
Combine and .
Step 13.2.5
Move to the left of .
Step 13.2.6
Cancel the common factor of .
Step 13.2.6.1
Cancel the common factor.
Step 13.2.6.2
Divide by .
Step 14
Replace all occurrences of with .
Step 15
Step 15.1
Simplify the numerator.
Step 15.1.1
Apply the distributive property.
Step 15.1.2
Combine and .
Step 15.1.3
Combine and .
Step 15.2
Factor out of .
Step 15.3
Factor out of .
Step 15.4
Factor out of .
Step 15.5
Factor out of .
Step 15.6
Factor out of .
Step 15.7
Rewrite as .
Step 15.8
Move the negative in front of the fraction.
Step 15.9
Move the negative in front of the fraction.
Step 15.10
Move the negative in front of the fraction.
Step 15.11
Reorder terms.