Last updated Unknown

Status: Tags: Links: Integral Calculus Notes

Integrals of Rational Functions

Check List

1. Polynomial long division
2. Try direct substitution
3. Try to manipulate the numerator to split the fraction
4. If denominator is a factorable quadratic, completing the square

Tips

Reminders

Frequent Mistakes

• Arithmetic mistakes when solving for constants

Techniques of Integration

• Partial Fraction Decomposition

Long Division

• Should be used when integrand is a rational function and the degree of the numerator >= the degree of the denominator
• Be sure to change signs before calculating the remainder for the next layer

Practice

Section D Part I

Completing the Square

• Used after long division when the denominator is a quadratic that can’t be factored and directly substituted -ex) $\frac{1}{x^2+4x+5}$

Steps

• Take half of b and square it
• add/subtract the difference of c from the calculated value above
• Create a binomial square and add the difference at the end
• Use u substitution with u as the binomial square
• Should have a tan inverse at the end

Practice

Section D Part II

Splitting the Fraction

• Used when long division and u substitution does not work

Steps

1. Make numerator suitable for u substitution by adding an extra constant
2. Split the fraction into one to use u substitution on, then one to use completing the square on
3. Solve both integrals

ANKI

START Cloze Integrating Rational Functions: Step 1: Try {Long Division} Step 2: Try {Direct Substitution} Step 3: If denominator is factorable, use {Partial Fraction Decomposition} If denominator is not factorable, use {Completing the Square, then Split the Fraction::2 Methods}. Integrate using {DS and Tan Inverse::2 Methods} Back: DECK: IntegralCalculus Tags: RationalIntegrals

END

START Cloze Polynomial Long Division: Dividend is the {numerator} Divisor is the {denominator} Quotient is {integrable} Remainder stays {divided by the denominator} Back: DECK: IntegralCalculus Tags: RationalIntegrals

END

START Cloze Completing the Square:

1. B is {multipled by 2 and squared}
2. Subtract c from {value above}
3. Create a binomial square and add difference at the end
4. U substitution with u as {the binomial square} Back: DECK: IntegralCalculus Tags: RationalIntegrals

END

START Cloze Splitting the Fraction:

1. Make numerator suitable for u substitution by {adding an extra constant} Divisor is the {denominator}
2. Split the fraction into one to use {u substitution} on, then one to use {completing the square} on
3. Solve both integrals Back: DECK: IntegralCalculus Tags: RationalIntegrals

END

Practice

Section D Part III

References: