Calculus Examples

Find the Integral cos(x)^2-sin(x)^2
Step 1
Split the single integral into multiple integrals.
Step 2
Use the half-angle formula to rewrite as .
Step 3
Since is constant with respect to , move out of the integral.
Step 4
Split the single integral into multiple integrals.
Step 5
Apply the constant rule.
Step 6
Let . Then , so . Rewrite using and .
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Step 6.1
Let . Find .
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Step 6.1.1
Differentiate .
Step 6.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 6.1.3
Differentiate using the Power Rule which states that is where .
Step 6.1.4
Multiply by .
Step 6.2
Rewrite the problem using and .
Step 7
Combine and .
Step 8
Since is constant with respect to , move out of the integral.
Step 9
The integral of with respect to is .
Step 10
Since is constant with respect to , move out of the integral.
Step 11
Use the half-angle formula to rewrite as .
Step 12
Since is constant with respect to , move out of the integral.
Step 13
Split the single integral into multiple integrals.
Step 14
Apply the constant rule.
Step 15
Since is constant with respect to , move out of the integral.
Step 16
Let . Then , so . Rewrite using and .
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Step 16.1
Let . Find .
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Step 16.1.1
Differentiate .
Step 16.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 16.1.3
Differentiate using the Power Rule which states that is where .
Step 16.1.4
Multiply by .
Step 16.2
Rewrite the problem using and .
Step 17
Combine and .
Step 18
Since is constant with respect to , move out of the integral.
Step 19
The integral of with respect to is .
Step 20
Simplify.
Step 21
Substitute back in for each integration substitution variable.
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Step 21.1
Replace all occurrences of with .
Step 21.2
Replace all occurrences of with .
Step 22
Simplify.
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Step 22.1
Combine and .
Step 22.2
Apply the distributive property.
Step 22.3
Combine and .
Step 22.4
Multiply .
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Step 22.4.1
Multiply by .
Step 22.4.2
Multiply by .
Step 22.4.3
Multiply by .
Step 22.4.4
Multiply by .
Step 22.5
Combine and .
Step 22.6
Apply the distributive property.
Step 22.7
Combine and .
Step 22.8
Multiply .
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Step 22.8.1
Multiply by .
Step 22.8.2
Multiply by .
Step 23
Reorder terms.