Calculus Examples

Evaluate the Integral integral of sin(pix)^2cos(pix)^5 with respect to x
Step 1
Let . Then , so . Rewrite using and .
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Step 1.1
Let . Find .
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Step 1.1.1
Differentiate .
Step 1.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.3
Differentiate using the Power Rule which states that is where .
Step 1.1.4
Multiply by .
Step 1.2
Rewrite the problem using and .
Step 2
Simplify.
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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
Factor out .
Step 5
Simplify with factoring out.
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Step 5.1
Factor out of .
Step 5.2
Rewrite as exponentiation.
Step 6
Using the Pythagorean Identity, rewrite as .
Step 7
Let . Then , so . 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
The derivative of with respect to is .
Step 7.2
Rewrite the problem using and .
Step 8
Expand .
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Step 8.1
Rewrite as .
Step 8.2
Apply the distributive property.
Step 8.3
Apply the distributive property.
Step 8.4
Apply the distributive property.
Step 8.5
Apply the distributive property.
Step 8.6
Apply the distributive property.
Step 8.7
Apply the distributive property.
Step 8.8
Move .
Step 8.9
Move parentheses.
Step 8.10
Move .
Step 8.11
Move .
Step 8.12
Move parentheses.
Step 8.13
Move .
Step 8.14
Move .
Step 8.15
Move parentheses.
Step 8.16
Move parentheses.
Step 8.17
Move .
Step 8.18
Multiply by .
Step 8.19
Multiply by .
Step 8.20
Multiply by .
Step 8.21
Factor out negative.
Step 8.22
Use the power rule to combine exponents.
Step 8.23
Add and .
Step 8.24
Multiply by .
Step 8.25
Factor out negative.
Step 8.26
Use the power rule to combine exponents.
Step 8.27
Add and .
Step 8.28
Multiply by .
Step 8.29
Multiply by .
Step 8.30
Use the power rule to combine exponents.
Step 8.31
Add and .
Step 8.32
Use the power rule to combine exponents.
Step 8.33
Add and .
Step 8.34
Subtract from .
Step 8.35
Reorder and .
Step 8.36
Move .
Step 9
Split the single integral into multiple integrals.
Step 10
By the Power Rule, the integral of with respect to is .
Step 11
Since is constant with respect to , move out of the integral.
Step 12
By the Power Rule, the integral of with respect to is .
Step 13
By the Power Rule, the integral of with respect to is .
Step 14
Simplify.
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Step 14.1
Simplify.
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Step 14.1.1
Combine and .
Step 14.1.2
Combine and .
Step 14.1.3
Combine and .
Step 14.2
Simplify.
Step 15
Substitute back in for each integration substitution variable.
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Step 15.1
Replace all occurrences of with .
Step 15.2
Replace all occurrences of with .
Step 16
Reorder terms.