Calculus Examples

Evaluate the Integral integral of sin(x)^4 with respect to x
Step 1
Simplify with factoring out.
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Step 1.1
Factor out of .
Step 1.2
Rewrite as exponentiation.
Step 2
Use the half-angle formula to rewrite as .
Step 3
Let . Then , so . Rewrite using and .
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Step 3.1
Let . Find .
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Step 3.1.1
Differentiate .
Step 3.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 3.1.3
Differentiate using the Power Rule which states that is where .
Step 3.1.4
Multiply by .
Step 3.2
Rewrite the problem using and .
Step 4
Since is constant with respect to , move out of the integral.
Step 5
Simplify by multiplying through.
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Step 5.1
Rewrite as a product.
Step 5.2
Expand .
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Step 5.2.1
Rewrite the exponentiation as a product.
Step 5.2.2
Apply the distributive property.
Step 5.2.3
Apply the distributive property.
Step 5.2.4
Apply the distributive property.
Step 5.2.5
Apply the distributive property.
Step 5.2.6
Apply the distributive property.
Step 5.2.7
Reorder and .
Step 5.2.8
Reorder and .
Step 5.2.9
Move .
Step 5.2.10
Reorder and .
Step 5.2.11
Reorder and .
Step 5.2.12
Move parentheses.
Step 5.2.13
Move .
Step 5.2.14
Reorder and .
Step 5.2.15
Reorder and .
Step 5.2.16
Move .
Step 5.2.17
Move .
Step 5.2.18
Reorder and .
Step 5.2.19
Reorder and .
Step 5.2.20
Move parentheses.
Step 5.2.21
Move .
Step 5.2.22
Move .
Step 5.2.23
Multiply by .
Step 5.2.24
Multiply by .
Step 5.2.25
Multiply by .
Step 5.2.26
Multiply by .
Step 5.2.27
Multiply by .
Step 5.2.28
Combine and .
Step 5.2.29
Multiply by .
Step 5.2.30
Combine and .
Step 5.2.31
Multiply by .
Step 5.2.32
Combine and .
Step 5.2.33
Combine and .
Step 5.2.34
Multiply by .
Step 5.2.35
Multiply by .
Step 5.2.36
Multiply by .
Step 5.2.37
Combine and .
Step 5.2.38
Multiply by .
Step 5.2.39
Multiply by .
Step 5.2.40
Combine and .
Step 5.2.41
Raise to the power of .
Step 5.2.42
Raise to the power of .
Step 5.2.43
Use the power rule to combine exponents.
Step 5.2.44
Add and .
Step 5.2.45
Subtract from .
Step 5.2.46
Combine and .
Step 5.2.47
Reorder and .
Step 5.2.48
Reorder and .
Step 5.3
Simplify.
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Step 5.3.1
Cancel the common factor of and .
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Step 5.3.1.1
Factor out of .
Step 5.3.1.2
Cancel the common factors.
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Step 5.3.1.2.1
Factor out of .
Step 5.3.1.2.2
Cancel the common factor.
Step 5.3.1.2.3
Rewrite the expression.
Step 5.3.2
Move the negative in front of the fraction.
Step 6
Split the single integral into multiple integrals.
Step 7
Since is constant with respect to , move out of the integral.
Step 8
Use the half-angle formula to rewrite as .
Step 9
Since is constant with respect to , move out of the integral.
Step 10
Simplify.
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Step 10.1
Multiply by .
Step 10.2
Multiply by .
Step 11
Split the single integral into multiple integrals.
Step 12
Apply the constant rule.
Step 13
Let . Then , so . Rewrite using and .
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Step 13.1
Let . Find .
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Step 13.1.1
Differentiate .
Step 13.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 13.1.3
Differentiate using the Power Rule which states that is where .
Step 13.1.4
Multiply by .
Step 13.2
Rewrite the problem using and .
Step 14
Combine and .
Step 15
Since is constant with respect to , move out of the integral.
Step 16
The integral of with respect to is .
Step 17
Apply the constant rule.
Step 18
Combine and .
Step 19
Since is constant with respect to , move out of the integral.
Step 20
Since is constant with respect to , move out of the integral.
Step 21
The integral of with respect to is .
Step 22
Simplify.
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Step 22.1
Simplify.
Step 22.2
Simplify.
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Step 22.2.1
To write as a fraction with a common denominator, multiply by .
Step 22.2.2
Write each expression with a common denominator of , by multiplying each by an appropriate factor of .
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Step 22.2.2.1
Multiply by .
Step 22.2.2.2
Multiply by .
Step 22.2.3
Combine the numerators over the common denominator.
Step 22.2.4
Move to the left of .
Step 22.2.5
Add and .
Step 23
Substitute back in for each integration substitution variable.
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Step 23.1
Replace all occurrences of with .
Step 23.2
Replace all occurrences of with .
Step 23.3
Replace all occurrences of with .
Step 24
Simplify.
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Step 24.1
Simplify each term.
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Step 24.1.1
Cancel the common factor of and .
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Step 24.1.1.1
Factor out of .
Step 24.1.1.2
Cancel the common factors.
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Step 24.1.1.2.1
Factor out of .
Step 24.1.1.2.2
Cancel the common factor.
Step 24.1.1.2.3
Rewrite the expression.
Step 24.1.2
Multiply by .
Step 24.2
Apply the distributive property.
Step 24.3
Simplify.
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Step 24.3.1
Multiply .
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Step 24.3.1.1
Multiply by .
Step 24.3.1.2
Multiply by .
Step 24.3.2
Multiply .
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Step 24.3.2.1
Multiply by .
Step 24.3.2.2
Multiply by .
Step 24.3.3
Multiply .
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Step 24.3.3.1
Multiply by .
Step 24.3.3.2
Multiply by .
Step 25
Reorder terms.