Calculus Examples

Evaluate the Integral integral of sin(4x)^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
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
Since is constant with respect to , move out of the integral.
Step 9
Expand .
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Step 9.1
Rewrite as .
Step 9.2
Apply the distributive property.
Step 9.3
Apply the distributive property.
Step 9.4
Apply the distributive property.
Step 9.5
Move .
Step 9.6
Move .
Step 9.7
Multiply by .
Step 9.8
Multiply by .
Step 9.9
Multiply by .
Step 9.10
Multiply by .
Step 9.11
Multiply by .
Step 9.12
Use the power rule to combine exponents.
Step 9.13
Add and .
Step 9.14
Subtract from .
Step 9.15
Reorder and .
Step 9.16
Move .
Step 10
Split the single integral into multiple integrals.
Step 11
By the Power Rule, the integral of with respect to is .
Step 12
Since is constant with respect to , move out of the integral.
Step 13
By the Power Rule, the integral of with respect to is .
Step 14
Apply the constant rule.
Step 15
Simplify.
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Step 15.1
Simplify.
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Step 15.1.1
Combine and .
Step 15.1.2
Combine and .
Step 15.2
Simplify.
Step 16
Substitute back in for each integration substitution variable.
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Step 16.1
Replace all occurrences of with .
Step 16.2
Replace all occurrences of with .
Step 17
Reorder terms.