Calculus Examples

Evaluate the Integral integral from 0 to 1 of (e^(2x)-e^(-2x))/(e^(2x)+e^(-2x)) with respect to x
Step 1
Let . Then , so . Rewrite using and .
Tap for more steps...
Step 1.1
Let . Find .
Tap for more steps...
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 .
Tap for more steps...
Step 1.1.3.1
Differentiate using the chain rule, which states that is where and .
Tap for more steps...
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 .
Tap for more steps...
Step 1.1.4.1
Differentiate using the chain rule, which states that is where and .
Tap for more steps...
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.
Tap for more steps...
Step 1.3.1
Simplify each term.
Tap for more steps...
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.
Tap for more steps...
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
Simplify.
Tap for more steps...
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
Simplify.
Tap for more steps...
Step 6.1
Use the quotient property of logarithms, .
Step 6.2
Combine and .
Step 7
Simplify.
Tap for more steps...
Step 7.1
Simplify the numerator.
Tap for more steps...
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.
Tap for more steps...
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