Calculus Examples

Evaluate the Integral integral of 1/(1+cos(x)) with respect to x
Step 1
Use the double-angle identity to transform to .
Step 2
Use the pythagorean identity to transform to .
Step 3
Simplify.
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Step 3.1
Subtract from .
Step 3.2
Add and .
Step 3.3
Add and .
Step 4
Multiply the argument by
Step 5
Combine.
Step 6
Multiply by .
Step 7
Rewrite in terms of sines and cosines.
Step 8
Reduce the expression by cancelling the common factors.
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Step 8.1
Apply the product rule to .
Step 8.2
Cancel the common factor of .
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Step 8.2.1
Factor out of .
Step 8.2.2
Cancel the common factor.
Step 8.2.3
Rewrite the expression.
Step 8.3
Simplify the expression.
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Step 8.3.1
One to any power is one.
Step 8.3.2
Multiply by .
Step 9
Since is constant with respect to , move out of the integral.
Step 10
Let . Then , so . Rewrite using and .
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Step 10.1
Let . Find .
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Step 10.1.1
Differentiate .
Step 10.1.2
Since is constant with respect to , the derivative of with respect to is .
Step 10.1.3
Differentiate using the Power Rule which states that is where .
Step 10.1.4
Multiply by .
Step 10.2
Rewrite the problem using and .
Step 11
Simplify.
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Step 11.1
Multiply by the reciprocal of the fraction to divide by .
Step 11.2
Multiply by .
Step 11.3
Move to the left of .
Step 12
Since is constant with respect to , move out of the integral.
Step 13
Simplify.
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Step 13.1
Combine and .
Step 13.2
Cancel the common factor of .
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Step 13.2.1
Cancel the common factor.
Step 13.2.2
Rewrite the expression.
Step 13.3
Multiply by .
Step 14
Since the derivative of is , the integral of is .
Step 15
Replace all occurrences of with .
Step 16
Reorder terms.