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Calculus Examples
Step 1
Step 1.1
Multiply both sides by .
Step 1.2
Cancel the common factor of .
Step 1.2.1
Cancel the common factor.
Step 1.2.2
Rewrite the expression.
Step 1.3
Remove unnecessary parentheses.
Step 1.4
Rewrite the equation.
Step 2
Step 2.1
Set up an integral on each side.
Step 2.2
Integrate the left side.
Step 2.2.1
Simplify.
Step 2.2.1.1
Combine and .
Step 2.2.1.2
Multiply by .
Step 2.2.1.3
Move to the left of .
Step 2.2.2
Since is constant with respect to , move out of the integral.
Step 2.2.3
Let . Then , so . Rewrite using and .
Step 2.2.3.1
Let . Find .
Step 2.2.3.1.1
Differentiate .
Step 2.2.3.1.2
Differentiate.
Step 2.2.3.1.2.1
By the Sum Rule, the derivative of with respect to is .
Step 2.2.3.1.2.2
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.3.1.3
Evaluate .
Step 2.2.3.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 2.2.3.1.3.2
Differentiate using the Power Rule which states that is where .
Step 2.2.3.1.3.3
Multiply by .
Step 2.2.3.1.4
Subtract from .
Step 2.2.3.2
Rewrite the problem using and .
Step 2.2.4
Simplify.
Step 2.2.4.1
Move the negative in front of the fraction.
Step 2.2.4.2
Dividing two negative values results in a positive value.
Step 2.2.4.3
Multiply by the reciprocal of the fraction to divide by .
Step 2.2.4.4
Multiply by .
Step 2.2.4.5
Multiply by .
Step 2.2.4.6
Combine and .
Step 2.2.4.7
Move the negative in front of the fraction.
Step 2.2.5
Since is constant with respect to , move out of the integral.
Step 2.2.6
Since is constant with respect to , move out of the integral.
Step 2.2.7
Simplify.
Step 2.2.7.1
Multiply by .
Step 2.2.7.2
Combine and .
Step 2.2.7.3
Multiply by .
Step 2.2.7.4
Cancel the common factor of and .
Step 2.2.7.4.1
Factor out of .
Step 2.2.7.4.2
Cancel the common factors.
Step 2.2.7.4.2.1
Factor out of .
Step 2.2.7.4.2.2
Cancel the common factor.
Step 2.2.7.4.2.3
Rewrite the expression.
Step 2.2.7.4.2.4
Divide by .
Step 2.2.8
The integral of with respect to is .
Step 2.2.9
Simplify.
Step 2.2.10
Replace all occurrences of with .
Step 2.2.11
Reorder terms.
Step 2.3
Apply the constant rule.
Step 2.4
Group the constant of integration on the right side as .
Step 3
Step 3.1
Divide each term in by and simplify.
Step 3.1.1
Divide each term in by .
Step 3.1.2
Simplify the left side.
Step 3.1.2.1
Cancel the common factor of .
Step 3.1.2.1.1
Cancel the common factor.
Step 3.1.2.1.2
Divide by .
Step 3.1.2.2
Combine and .
Step 3.1.3
Simplify the right side.
Step 3.1.3.1
Simplify each term.
Step 3.1.3.1.1
Move the negative in front of the fraction.
Step 3.1.3.1.2
Move the negative in front of the fraction.
Step 3.2
To solve for , rewrite the equation using properties of logarithms.
Step 3.3
Rewrite in exponential form using the definition of a logarithm. If and are positive real numbers and , then is equivalent to .
Step 3.4
Solve for .
Step 3.4.1
Rewrite the equation as .
Step 3.4.2
Remove the absolute value term. This creates a on the right side of the equation because .
Step 3.4.3
Subtract from both sides of the equation.
Step 3.4.4
Multiply both sides of the equation by .
Step 3.4.5
Simplify both sides of the equation.
Step 3.4.5.1
Simplify the left side.
Step 3.4.5.1.1
Simplify .
Step 3.4.5.1.1.1
Cancel the common factor of .
Step 3.4.5.1.1.1.1
Move the leading negative in into the numerator.
Step 3.4.5.1.1.1.2
Factor out of .
Step 3.4.5.1.1.1.3
Cancel the common factor.
Step 3.4.5.1.1.1.4
Rewrite the expression.
Step 3.4.5.1.1.2
Multiply.
Step 3.4.5.1.1.2.1
Multiply by .
Step 3.4.5.1.1.2.2
Multiply by .
Step 3.4.5.2
Simplify the right side.
Step 3.4.5.2.1
Simplify .
Step 3.4.5.2.1.1
Apply the distributive property.
Step 3.4.5.2.1.2
Multiply by .
Step 3.4.6
Reorder and .
Step 4
Step 4.1
Simplify the constant of integration.
Step 4.2
Reorder terms.
Step 4.3
Rewrite as .
Step 4.4
Reorder and .
Step 4.5
Combine constants with the plus or minus.