Calculus Examples

Evaluate the Integral integral from 0 to 1 of 2/(2x^2+3x+1) with respect to x
Step 1
Since is constant with respect to , move out of the integral.
Step 2
Write the fraction using partial fraction decomposition.
Tap for more steps...
Step 2.1
Decompose the fraction and multiply through by the common denominator.
Tap for more steps...
Step 2.1.1
Factor by grouping.
Tap for more steps...
Step 2.1.1.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Tap for more steps...
Step 2.1.1.1.1
Factor out of .
Step 2.1.1.1.2
Rewrite as plus
Step 2.1.1.1.3
Apply the distributive property.
Step 2.1.1.1.4
Multiply by .
Step 2.1.1.2
Factor out the greatest common factor from each group.
Tap for more steps...
Step 2.1.1.2.1
Group the first two terms and the last two terms.
Step 2.1.1.2.2
Factor out the greatest common factor (GCF) from each group.
Step 2.1.1.3
Factor the polynomial by factoring out the greatest common factor, .
Step 2.1.2
For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place .
Step 2.1.3
For each factor in the denominator, create a new fraction using the factor as the denominator, and an unknown value as the numerator. Since the factor in the denominator is linear, put a single variable in its place .
Step 2.1.4
Multiply each fraction in the equation by the denominator of the original expression. In this case, the denominator is .
Step 2.1.5
Cancel the common factor of .
Tap for more steps...
Step 2.1.5.1
Cancel the common factor.
Step 2.1.5.2
Rewrite the expression.
Step 2.1.6
Cancel the common factor of .
Tap for more steps...
Step 2.1.6.1
Cancel the common factor.
Step 2.1.6.2
Rewrite the expression.
Step 2.1.7
Simplify each term.
Tap for more steps...
Step 2.1.7.1
Cancel the common factor of .
Tap for more steps...
Step 2.1.7.1.1
Cancel the common factor.
Step 2.1.7.1.2
Divide by .
Step 2.1.7.2
Apply the distributive property.
Step 2.1.7.3
Multiply by .
Step 2.1.7.4
Cancel the common factor of .
Tap for more steps...
Step 2.1.7.4.1
Cancel the common factor.
Step 2.1.7.4.2
Divide by .
Step 2.1.7.5
Apply the distributive property.
Step 2.1.7.6
Rewrite using the commutative property of multiplication.
Step 2.1.7.7
Multiply by .
Step 2.1.8
Simplify the expression.
Tap for more steps...
Step 2.1.8.1
Move .
Step 2.1.8.2
Move .
Step 2.2
Create equations for the partial fraction variables and use them to set up a system of equations.
Tap for more steps...
Step 2.2.1
Create an equation for the partial fraction variables by equating the coefficients of from each side of the equation. For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.
Step 2.2.2
Create an equation for the partial fraction variables by equating the coefficients of the terms not containing . For the equation to be equal, the equivalent coefficients on each side of the equation must be equal.
Step 2.2.3
Set up the system of equations to find the coefficients of the partial fractions.
Step 2.3
Solve the system of equations.
Tap for more steps...
Step 2.3.1
Solve for in .
Tap for more steps...
Step 2.3.1.1
Rewrite the equation as .
Step 2.3.1.2
Subtract from both sides of the equation.
Step 2.3.2
Replace all occurrences of with in each equation.
Tap for more steps...
Step 2.3.2.1
Replace all occurrences of in with .
Step 2.3.2.2
Simplify the right side.
Tap for more steps...
Step 2.3.2.2.1
Add and .
Step 2.3.3
Solve for in .
Tap for more steps...
Step 2.3.3.1
Rewrite the equation as .
Step 2.3.3.2
Divide each term in by and simplify.
Tap for more steps...
Step 2.3.3.2.1
Divide each term in by .
Step 2.3.3.2.2
Simplify the left side.
Tap for more steps...
Step 2.3.3.2.2.1
Dividing two negative values results in a positive value.
Step 2.3.3.2.2.2
Divide by .
Step 2.3.3.2.3
Simplify the right side.
Tap for more steps...
Step 2.3.3.2.3.1
Divide by .
Step 2.3.4
Replace all occurrences of with in each equation.
Tap for more steps...
Step 2.3.4.1
Replace all occurrences of in with .
Step 2.3.4.2
Simplify the right side.
Tap for more steps...
Step 2.3.4.2.1
Multiply by .
Step 2.3.5
List all of the solutions.
Step 2.4
Replace each of the partial fraction coefficients in with the values found for and .
Step 2.5
Move the negative in front of the fraction.
Step 3
Split the single integral into multiple integrals.
Step 4
Since is constant with respect to , move out of the integral.
Step 5
Let . Then , so . Rewrite using and .
Tap for more steps...
Step 5.1
Let . Find .
Tap for more steps...
Step 5.1.1
Differentiate .
Step 5.1.2
By the Sum Rule, the derivative of with respect to is .
Step 5.1.3
Evaluate .
Tap for more steps...
Step 5.1.3.1
Since is constant with respect to , the derivative of with respect to is .
Step 5.1.3.2
Differentiate using the Power Rule which states that is where .
Step 5.1.3.3
Multiply by .
Step 5.1.4
Differentiate using the Constant Rule.
Tap for more steps...
Step 5.1.4.1
Since is constant with respect to , the derivative of with respect to is .
Step 5.1.4.2
Add and .
Step 5.2
Substitute the lower limit in for in .
Step 5.3
Simplify.
Tap for more steps...
Step 5.3.1
Multiply by .
Step 5.3.2
Add and .
Step 5.4
Substitute the upper limit in for in .
Step 5.5
Simplify.
Tap for more steps...
Step 5.5.1
Multiply by .
Step 5.5.2
Add and .
Step 5.6
The values found for and will be used to evaluate the definite integral.
Step 5.7
Rewrite the problem using , , and the new limits of integration.
Step 6
Simplify.
Tap for more steps...
Step 6.1
Multiply by .
Step 6.2
Move to the left of .
Step 7
Since is constant with respect to , move out of the integral.
Step 8
Simplify.
Tap for more steps...
Step 8.1
Combine and .
Step 8.2
Cancel the common factor of .
Tap for more steps...
Step 8.2.1
Cancel the common factor.
Step 8.2.2
Rewrite the expression.
Step 8.3
Multiply by .
Step 9
The integral of with respect to is .
Step 10
Since is constant with respect to , move out of the integral.
Step 11
Let . Then . Rewrite using and .
Tap for more steps...
Step 11.1
Let . Find .
Tap for more steps...
Step 11.1.1
Differentiate .
Step 11.1.2
By the Sum Rule, the derivative of with respect to is .
Step 11.1.3
Differentiate using the Power Rule which states that is where .
Step 11.1.4
Since is constant with respect to , the derivative of with respect to is .
Step 11.1.5
Add and .
Step 11.2
Substitute the lower limit in for in .
Step 11.3
Add and .
Step 11.4
Substitute the upper limit in for in .
Step 11.5
Add and .
Step 11.6
The values found for and will be used to evaluate the definite integral.
Step 11.7
Rewrite the problem using , , and the new limits of integration.
Step 12
The integral of with respect to is .
Step 13
Substitute and simplify.
Tap for more steps...
Step 13.1
Evaluate at and at .
Step 13.2
Evaluate at and at .
Step 13.3
Remove parentheses.
Step 14
Simplify.
Tap for more steps...
Step 14.1
Use the quotient property of logarithms, .
Step 14.2
Use the quotient property of logarithms, .
Step 14.3
Use the quotient property of logarithms, .
Step 14.4
Rewrite as a product.
Step 14.5
Multiply by the reciprocal of the fraction to divide by .
Step 14.6
Multiply by .
Step 14.7
Multiply by .
Step 14.8
To multiply absolute values, multiply the terms inside each absolute value.
Step 14.9
Multiply by .
Step 14.10
To multiply absolute values, multiply the terms inside each absolute value.
Step 14.11
Multiply by .
Step 15
Simplify.
Tap for more steps...
Step 15.1
The absolute value is the distance between a number and zero. The distance between and is .
Step 15.2
The absolute value is the distance between a number and zero. The distance between and is .
Step 16
The result can be shown in multiple forms.
Exact Form:
Decimal Form:
Step 17