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Calculus Examples
Step 1
Step 1.1
Let . Find .
Step 1.1.1
Differentiate .
Step 1.1.2
By the Sum Rule, the derivative of with respect to is .
Step 1.1.3
Differentiate using the Power Rule which states that is where .
Step 1.1.4
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.5
Add and .
Step 1.2
Substitute the lower limit in for in .
Step 1.3
Add and .
Step 1.4
Substitute the upper limit in for in .
Step 1.5
Add and .
Step 1.6
The values found for and will be used to evaluate the definite integral.
Step 1.7
Rewrite the problem using , , and the new limits of integration.
Step 2
Use to rewrite as .
Step 3
By the Power Rule, the integral of with respect to is .
Step 4
Step 4.1
Evaluate at and at .
Step 4.2
Simplify.
Step 4.2.1
Rewrite as .
Step 4.2.2
Apply the power rule and multiply exponents, .
Step 4.2.3
Cancel the common factor of .
Step 4.2.3.1
Cancel the common factor.
Step 4.2.3.2
Rewrite the expression.
Step 4.2.4
Raise to the power of .
Step 4.2.5
Combine and .
Step 4.2.6
Multiply by .
Step 4.2.7
Cancel the common factor of and .
Step 4.2.7.1
Factor out of .
Step 4.2.7.2
Cancel the common factors.
Step 4.2.7.2.1
Factor out of .
Step 4.2.7.2.2
Cancel the common factor.
Step 4.2.7.2.3
Rewrite the expression.
Step 4.2.7.2.4
Divide by .
Step 4.2.8
Rewrite as .
Step 4.2.9
Apply the power rule and multiply exponents, .
Step 4.2.10
Cancel the common factor of .
Step 4.2.10.1
Cancel the common factor.
Step 4.2.10.2
Rewrite the expression.
Step 4.2.11
Raise to the power of .
Step 4.2.12
Multiply by .
Step 4.2.13
Combine and .
Step 4.2.14
Multiply by .
Step 4.2.15
Move the negative in front of the fraction.
Step 4.2.16
To write as a fraction with a common denominator, multiply by .
Step 4.2.17
Combine and .
Step 4.2.18
Combine the numerators over the common denominator.
Step 4.2.19
Simplify the numerator.
Step 4.2.19.1
Multiply by .
Step 4.2.19.2
Subtract from .
Step 5
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Mixed Number Form:
Step 6