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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
Evaluate .
Step 1.1.3.1
Differentiate using the chain rule, which states that is where and .
Step 1.1.3.1.1
To apply the Chain Rule, set as .
Step 1.1.3.1.2
Differentiate using the Exponential Rule which states that is where =.
Step 1.1.3.1.3
Replace all occurrences of with .
Step 1.1.3.2
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.3.3
Differentiate using the Power Rule which states that is where .
Step 1.1.3.4
Multiply by .
Step 1.1.3.5
Move to the left of .
Step 1.1.4
Evaluate .
Step 1.1.4.1
Differentiate using the chain rule, which states that is where and .
Step 1.1.4.1.1
To apply the Chain Rule, set as .
Step 1.1.4.1.2
Differentiate using the Exponential Rule which states that is where =.
Step 1.1.4.1.3
Replace all occurrences of with .
Step 1.1.4.2
Since is constant with respect to , the derivative of with respect to is .
Step 1.1.4.3
Differentiate using the Power Rule which states that is where .
Step 1.1.4.4
Multiply by .
Step 1.1.4.5
Move to the left of .
Step 1.2
Substitute the lower limit in for in .
Step 1.3
Simplify.
Step 1.3.1
Simplify each term.
Step 1.3.1.1
Multiply by .
Step 1.3.1.2
Anything raised to is .
Step 1.3.1.3
Multiply by .
Step 1.3.1.4
Anything raised to is .
Step 1.3.2
Add and .
Step 1.4
Substitute the upper limit in for in .
Step 1.5
Simplify each term.
Step 1.5.1
Multiply by .
Step 1.5.2
Multiply by .
Step 1.5.3
Rewrite the expression using the negative exponent rule .
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
Step 2.1
Multiply by .
Step 2.2
Move to the left of .
Step 3
Since is constant with respect to , move out of the integral.
Step 4
The integral of with respect to is .
Step 5
Evaluate at and at .
Step 6
Step 6.1
Use the quotient property of logarithms, .
Step 6.2
Combine and .
Step 7
Step 7.1
Simplify the numerator.
Step 7.1.1
is approximately which is positive so remove the absolute value
Step 7.1.2
To write as a fraction with a common denominator, multiply by .
Step 7.1.3
Combine the numerators over the common denominator.
Step 7.1.4
Multiply by by adding the exponents.
Step 7.1.4.1
Use the power rule to combine exponents.
Step 7.1.4.2
Add and .
Step 7.2
The absolute value is the distance between a number and zero. The distance between and is .
Step 7.3
Multiply the numerator by the reciprocal of the denominator.
Step 7.4
Multiply by .
Step 7.5
Move to the left of .
Step 8
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Step 9