Calculus Examples

Evaluate the Integral integral of sin(4x)cos(4x) 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
Differentiate using the chain rule, which states that is where and .
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Step 1.1.2.1
To apply the Chain Rule, set as .
Step 1.1.2.2
The derivative of with respect to is .
Step 1.1.2.3
Replace all occurrences of with .
Step 1.1.3
Differentiate.
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Step 1.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.3.2
Differentiate using the Power Rule which states that is where .
Step 1.1.3.3
Simplify the expression.
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Step 1.1.3.3.1
Multiply by .
Step 1.1.3.3.2
Move to the left of .
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
By the Power Rule, the integral of with respect to is .
Step 5
Simplify.
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Step 5.1
Rewrite as .
Step 5.2
Simplify.
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Step 5.2.1
Multiply by .
Step 5.2.2
Multiply by .
Step 6
Replace all occurrences of with .