Calculus Examples

Find the Integral tan(x)^6
Step 1
Simplify with factoring out.
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Step 1.1
Factor out of .
Step 1.2
Rewrite as exponentiation.
Step 2
Using the Pythagorean Identity, rewrite as .
Step 3
Simplify.
Step 4
Split the single integral into multiple integrals.
Step 5
Apply the constant rule.
Step 6
Since is constant with respect to , move out of the integral.
Step 7
Since the derivative of is , the integral of is .
Step 8
Since is constant with respect to , move out of the integral.
Step 9
Simplify the expression.
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Step 9.1
Rewrite as plus
Step 9.2
Rewrite as .
Step 10
Using the Pythagorean Identity, rewrite as .
Step 11
Let . Then , so . Rewrite using and .
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Step 11.1
Let . Find .
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Step 11.1.1
Differentiate .
Step 11.1.2
The derivative of with respect to is .
Step 11.2
Rewrite the problem using and .
Step 12
Split the single integral into multiple integrals.
Step 13
Apply the constant rule.
Step 14
By the Power Rule, the integral of with respect to is .
Step 15
Simplify with factoring out.
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Step 15.1
Combine and .
Step 15.2
Simplify the expression.
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Step 15.2.1
Rewrite as plus
Step 15.2.2
Rewrite as .
Step 15.3
Factor out of .
Step 15.4
Rewrite as exponentiation.
Step 16
Using the Pythagorean Identity, rewrite as .
Step 17
Let . Then , so . Rewrite using and .
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Step 17.1
Let . Find .
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Step 17.1.1
Differentiate .
Step 17.1.2
The derivative of with respect to is .
Step 17.2
Rewrite the problem using and .
Step 18
Expand .
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Step 18.1
Rewrite as .
Step 18.2
Apply the distributive property.
Step 18.3
Apply the distributive property.
Step 18.4
Apply the distributive property.
Step 18.5
Reorder and .
Step 18.6
Multiply by .
Step 18.7
Multiply by .
Step 18.8
Multiply by .
Step 18.9
Use the power rule to combine exponents.
Step 18.10
Add and .
Step 18.11
Add and .
Step 18.12
Reorder and .
Step 18.13
Move .
Step 19
Split the single integral into multiple integrals.
Step 20
By the Power Rule, the integral of with respect to is .
Step 21
Since is constant with respect to , move out of the integral.
Step 22
By the Power Rule, the integral of with respect to is .
Step 23
Combine and .
Step 24
Apply the constant rule.
Step 25
Simplify.
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Step 25.1
Simplify.
Step 25.2
Simplify.
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Step 25.2.1
To write as a fraction with a common denominator, multiply by .
Step 25.2.2
Combine and .
Step 25.2.3
Combine the numerators over the common denominator.
Step 25.2.4
Multiply by .
Step 25.2.5
Add and .
Step 25.2.6
Move the negative in front of the fraction.
Step 25.2.7
Add and .
Step 26
Substitute back in for each integration substitution variable.
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Step 26.1
Replace all occurrences of with .
Step 26.2
Replace all occurrences of with .
Step 27
Reorder terms.