Calculus Examples

Evaluate the Integral integral of sec(5x)^4 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
Simplify the expression.
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Step 4.1
Rewrite as plus
Step 4.2
Rewrite as .
Step 5
Using the Pythagorean Identity, rewrite as .
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
The derivative of with respect to is .
Step 6.2
Rewrite the problem using and .
Step 7
Split the single integral into multiple integrals.
Step 8
Apply the constant rule.
Step 9
By the Power Rule, the integral of with respect to is .
Step 10
Simplify.
Step 11
Substitute back in for each integration substitution variable.
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Step 11.1
Replace all occurrences of with .
Step 11.2
Replace all occurrences of with .
Step 12
Simplify.
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Step 12.1
Combine and .
Step 12.2
Apply the distributive property.
Step 12.3
Combine and .
Step 12.4
Combine.
Step 12.5
Simplify each term.
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Step 12.5.1
Multiply by .
Step 12.5.2
Multiply by .
Step 13
Reorder terms.