Calculus Examples

Evaluate the Integral integral of sec(x)^6 with respect to x
Step 1
Simplify with factoring out.
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Step 1.1
Rewrite as plus
Step 1.2
Rewrite as .
Step 1.3
Factor out of .
Step 1.4
Rewrite as exponentiation.
Step 2
Using the Pythagorean Identity, 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
The derivative of with respect to is .
Step 3.2
Rewrite the problem using and .
Step 4
Expand .
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Step 4.1
Rewrite as .
Step 4.2
Apply the distributive property.
Step 4.3
Apply the distributive property.
Step 4.4
Apply the distributive property.
Step 4.5
Reorder and .
Step 4.6
Multiply by .
Step 4.7
Multiply by .
Step 4.8
Multiply by .
Step 4.9
Use the power rule to combine exponents.
Step 4.10
Add and .
Step 4.11
Add and .
Step 4.12
Reorder and .
Step 4.13
Move .
Step 5
Split the single integral into multiple integrals.
Step 6
By the Power Rule, the integral of with respect to is .
Step 7
Since is constant with respect to , move out of the integral.
Step 8
By the Power Rule, the integral of with respect to is .
Step 9
Apply the constant rule.
Step 10
Simplify.
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Step 10.1
Combine and .
Step 10.2
Simplify.
Step 11
Replace all occurrences of with .
Step 12
Reorder terms.