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Calculus Examples
Step 1
Since is constant with respect to , move out of the integral.
Step 2
Step 2.1
Let . Find .
Step 2.1.1
Differentiate .
Step 2.1.2
By the Sum Rule, the derivative of with respect to is .
Step 2.1.3
Since is constant with respect to , the derivative of with respect to is .
Step 2.1.4
Differentiate using the Power Rule which states that is where .
Step 2.1.5
Add and .
Step 2.2
Substitute the lower limit in for in .
Step 2.3
Simplify.
Step 2.3.1
Raising to any positive power yields .
Step 2.3.2
Add and .
Step 2.4
Substitute the upper limit in for in .
Step 2.5
Simplify.
Step 2.5.1
Rewrite as .
Step 2.5.1.1
Use to rewrite as .
Step 2.5.1.2
Apply the power rule and multiply exponents, .
Step 2.5.1.3
Combine and .
Step 2.5.1.4
Cancel the common factor of .
Step 2.5.1.4.1
Cancel the common factor.
Step 2.5.1.4.2
Rewrite the expression.
Step 2.5.1.5
Evaluate the exponent.
Step 2.5.2
Add and .
Step 2.6
The values found for and will be used to evaluate the definite integral.
Step 2.7
Rewrite the problem using , , and the new limits of integration.
Step 3
Combine and .
Step 4
Since is constant with respect to , move out of the integral.
Step 5
Step 5.1
Combine and .
Step 5.2
Use to rewrite as .
Step 6
By the Power Rule, the integral of with respect to is .
Step 7
Step 7.1
Evaluate at and at .
Step 7.2
Simplify.
Step 7.2.1
Rewrite as .
Step 7.2.2
Apply the power rule and multiply exponents, .
Step 7.2.3
Cancel the common factor of .
Step 7.2.3.1
Cancel the common factor.
Step 7.2.3.2
Rewrite the expression.
Step 7.2.4
Raise to the power of .
Step 7.2.5
Combine and .
Step 7.2.6
Multiply by .
Step 7.2.7
Rewrite as .
Step 7.2.8
Apply the power rule and multiply exponents, .
Step 7.2.9
Cancel the common factor of .
Step 7.2.9.1
Cancel the common factor.
Step 7.2.9.2
Rewrite the expression.
Step 7.2.10
Raise to the power of .
Step 7.2.11
Multiply by .
Step 7.2.12
Combine and .
Step 7.2.13
Multiply by .
Step 7.2.14
Cancel the common factor of and .
Step 7.2.14.1
Factor out of .
Step 7.2.14.2
Cancel the common factors.
Step 7.2.14.2.1
Factor out of .
Step 7.2.14.2.2
Cancel the common factor.
Step 7.2.14.2.3
Rewrite the expression.
Step 7.2.14.2.4
Divide by .
Step 7.2.15
To write as a fraction with a common denominator, multiply by .
Step 7.2.16
Combine and .
Step 7.2.17
Combine the numerators over the common denominator.
Step 7.2.18
Simplify the numerator.
Step 7.2.18.1
Multiply by .
Step 7.2.18.2
Subtract from .
Step 7.2.19
Multiply by .
Step 7.2.20
Multiply by .
Step 7.2.21
Multiply by .
Step 8
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Mixed Number Form:
Step 9