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Calculus Examples
Step 1
Write the integral as a limit as approaches .
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
The values found for and will be used to evaluate the definite integral.
Step 2.6
Rewrite the problem using , , and the new limits of integration.
Step 3
Step 3.1
Multiply by .
Step 3.2
Move to the left of .
Step 4
Since is constant with respect to , move out of the integral.
Step 5
Step 5.1
Move out of the denominator by raising it to the power.
Step 5.2
Multiply the exponents in .
Step 5.2.1
Apply the power rule and multiply exponents, .
Step 5.2.2
Multiply by .
Step 6
By the Power Rule, the integral of with respect to is .
Step 7
Combine and .
Step 8
Step 8.1
Evaluate at and at .
Step 8.2
One to any power is one.
Step 9
Step 9.1
Factor out of .
Step 9.2
Rewrite as .
Step 9.3
Factor out of .
Step 9.4
Rewrite as .
Step 9.5
Move the negative in front of the fraction.
Step 10
Step 10.1
Move the term outside of the limit because it is constant with respect to .
Step 10.2
Move the term outside of the limit because it is constant with respect to .
Step 10.3
Split the limit using the Sum of Limits Rule on the limit as approaches .
Step 10.4
Rewrite the expression using the negative exponent rule .
Step 10.5
Since its numerator approaches a real number while its denominator is unbounded, the fraction approaches .
Step 10.6
Evaluate the limit.
Step 10.6.1
Evaluate the limit of which is constant as approaches .
Step 10.6.2
Simplify the answer.
Step 10.6.2.1
Multiply by .
Step 10.6.2.2
Subtract from .
Step 10.6.2.3
Multiply .
Step 10.6.2.3.1
Multiply by .
Step 10.6.2.3.2
Multiply by .
Step 11
The result can be shown in multiple forms.
Exact Form:
Decimal Form: