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Calculus Examples
Step 1
Let , where . Then . Note that since , is positive.
Step 2
Step 2.1
Simplify .
Step 2.1.1
Simplify each term.
Step 2.1.1.1
Apply the product rule to .
Step 2.1.1.2
Raise to the power of .
Step 2.1.2
Factor out of .
Step 2.1.2.1
Factor out of .
Step 2.1.2.2
Factor out of .
Step 2.1.2.3
Factor out of .
Step 2.1.3
Apply pythagorean identity.
Step 2.1.4
Rewrite as .
Step 2.1.5
Pull terms out from under the radical, assuming positive real numbers.
Step 2.2
Simplify.
Step 2.2.1
Multiply by .
Step 2.2.2
Multiply by by adding the exponents.
Step 2.2.2.1
Move .
Step 2.2.2.2
Multiply by .
Step 2.2.2.2.1
Raise to the power of .
Step 2.2.2.2.2
Use the power rule to combine exponents.
Step 2.2.2.3
Add and .
Step 3
Since is constant with respect to , move out of the integral.
Step 4
Factor out of .
Step 5
Integrate by parts using the formula , where and .
Step 6
Raise to the power of .
Step 7
Raise to the power of .
Step 8
Use the power rule to combine exponents.
Step 9
Step 9.1
Add and .
Step 9.2
Reorder and .
Step 10
Using the Pythagorean Identity, rewrite as .
Step 11
Step 11.1
Rewrite the exponentiation as a product.
Step 11.2
Apply the distributive property.
Step 11.3
Reorder and .
Step 12
Raise to the power of .
Step 13
Raise to the power of .
Step 14
Use the power rule to combine exponents.
Step 15
Add and .
Step 16
Raise to the power of .
Step 17
Use the power rule to combine exponents.
Step 18
Add and .
Step 19
Split the single integral into multiple integrals.
Step 20
Since is constant with respect to , move out of the integral.
Step 21
The integral of with respect to is .
Step 22
Step 22.1
Apply the distributive property.
Step 22.2
Multiply by .
Step 23
Solving for , we find that = .
Step 24
Multiply by .
Step 25
Simplify.
Step 26
Step 26.1
Combine and .
Step 26.2
Cancel the common factor of and .
Step 26.2.1
Factor out of .
Step 26.2.2
Cancel the common factors.
Step 26.2.2.1
Factor out of .
Step 26.2.2.2
Cancel the common factor.
Step 26.2.2.3
Rewrite the expression.
Step 26.2.2.4
Divide by .
Step 27
Replace all occurrences of with .
Step 28
Step 28.1
Simplify each term.
Step 28.1.1
Draw a triangle in the plane with vertices , , and the origin. Then is the angle between the positive x-axis and the ray beginning at the origin and passing through . Therefore, is .
Step 28.1.2
Apply the product rule to .
Step 28.1.3
Raise to the power of .
Step 28.1.4
Write as a fraction with a common denominator.
Step 28.1.5
Combine the numerators over the common denominator.
Step 28.1.6
Rewrite as .
Step 28.1.6.1
Factor the perfect power out of .
Step 28.1.6.2
Factor the perfect power out of .
Step 28.1.6.3
Rearrange the fraction .
Step 28.1.7
Pull terms out from under the radical.
Step 28.1.8
Combine and .
Step 28.1.9
The functions tangent and arctangent are inverses.
Step 28.1.10
Combine.
Step 28.1.11
Multiply by .
Step 28.1.12
Simplify each term.
Step 28.1.12.1
Draw a triangle in the plane with vertices , , and the origin. Then is the angle between the positive x-axis and the ray beginning at the origin and passing through . Therefore, is .
Step 28.1.12.2
Apply the product rule to .
Step 28.1.12.3
Raise to the power of .
Step 28.1.12.4
Write as a fraction with a common denominator.
Step 28.1.12.5
Combine the numerators over the common denominator.
Step 28.1.12.6
Rewrite as .
Step 28.1.12.6.1
Factor the perfect power out of .
Step 28.1.12.6.2
Factor the perfect power out of .
Step 28.1.12.6.3
Rearrange the fraction .
Step 28.1.12.7
Pull terms out from under the radical.
Step 28.1.12.8
Combine and .
Step 28.1.12.9
The functions tangent and arctangent are inverses.
Step 28.1.13
Combine the numerators over the common denominator.
Step 28.1.14
Remove non-negative terms from the absolute value.
Step 28.2
To write as a fraction with a common denominator, multiply by .
Step 28.3
Combine and .
Step 28.4
Combine the numerators over the common denominator.
Step 28.5
Move to the left of .
Step 28.6
Cancel the common factor of .
Step 28.6.1
Factor out of .
Step 28.6.2
Cancel the common factor.
Step 28.6.3
Rewrite the expression.
Step 29
Reorder terms.