Calculus with Fractions
\pi \cdot \pi
Calculus with Fractions
Adding/Subtracting Fractions
Add \(\frac{1}{2}+\frac{3}{5}\).
Write out all the steps in detail. Don't assume anything!
Calculus with Fractions
Adding/Subtracting Fractions
Are you missing any steps?
\(\frac{1}{2} + \frac{3}{5}\)
\(=\frac{1}{2}\cdot \frac{5}{5}+\frac{3}{5}\cdot \frac{2}{2}\)
\(=\frac{5}{10}+\frac{6}{10}\)
\(=\frac{5+6}{10}\)
\(=\frac{11}{10}\)
Calculus with Fractions
Adding/Subtracting Fractions
We can say that
\(\frac{1}{2} + \frac{3}{5}=\frac{11}{10}\)
and
\(\frac{11}{10}=\frac{1}{2} + \frac{3}{5}\).
Calculus with Fractions
Adding/Subtracting Fractions
Subtract \(\frac{1}{x+2}-\frac{3}{x-5}\).
Write out all the steps in detail. Don't assume anything!
Calculus with Fractions
Adding/Subtracting Fractions
Are you missing any steps?
\(\frac{1}{x+2}-\frac{3}{x-5}\)
\(=\frac{1}{x+2}\cdot \frac{x-5}{x-5}-\frac{3}{x-5}\cdot \frac{x+2}{x+2}\)
\(=\frac{1(x-5)}{(x+2)(x-5)}-\frac{3(x+2)}{(x-5)(x+2)}\)
\(=\frac{1(x-5)-3(x+2)}{(x+2)(x-5)}\)
\(=\frac{x-5-3x-6}{(x+2)(x-5)}\)
\(=\frac{-2x-11}{(x+2)(x-5)}\)
Calculus with Fractions
Adding/Subtracting Fractions
We can say that
\(\frac{1}{x+2}-\frac{3}{x-5}=\frac{-2x-11}{(x+2)(x-5)}\)
and
\(\frac{-2x-11}{(x+2)(x-5)}=\frac{1}{x+2}-\frac{3}{x-5}\).
Calculus with Fractions
Adding/Subtracting Fractions
What if you were given
\(\frac{-2x-11}{(x+2)(x-5)}\) and need to find that it came from \(\frac{1}{x+2}-\frac{3}{x-5}\)?
What would \(\frac{x+1}{(x-10)(x+4)}\) split into?
Calculus with Fractions
Integrating Fractions
What is \(\int \frac{1}{x+2}dx\)?
What is \(\int \frac{3}{x-5}dx\)?
What is \(\int \frac{1}{x+2}-\frac{3}{x-5}dx\)?
What is \(\int \frac{-2x-11}{(x+2)(x-5)}dx\)?
Calculus with Fractions
Integrating Fractions
If we can find a way to 'decompose' a complex fraction into its component 'partial fractions,' we can integrate those smaller fractions much more quickly.
Enter Partial Fraction Decomposition...
Calculus with Fractions
Classifying Problems
The denominator in \(\frac{3 x+11}{(x-3)(x+2)}\) has:
A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors
Calculus with Fractions
Classifying Problems
The denominator in \(\frac{3 x+11}{(x-3)(x+2)}\) has:
A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors
Calculus with Fractions
Classifying Problems
The denominator in \(\frac{2+x^4}{x(x^2+9)}\) has:
A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors
Calculus with Fractions
Classifying Problems
The denominator in \(\frac{2+x^4}{x(x^2+9)}\) has:
A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors
Calculus with Fractions
Classifying Problems
The denominator in \(\frac{3 x+11}{(x+2)(x+2)}\) has:
A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors
Calculus with Fractions
Classifying Problems
The denominator in \(\frac{3 x+11}{(x+2)(x+2)}\) has:
A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors
Calculus with Fractions
Classifying Problems
The denominator in \(\frac{3 x+11}{x^2+6x+9}\) has:
A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors
Calculus with Fractions
Classifying Problems
The denominator in \(\frac{3 x+11}{x^2+6x+9}\) has:
A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors
Calculus with Fractions
Classifying Problems
The denominator in \(\frac{4 x-11}{x^3-9 x^2}\) has:
A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors
Calculus with Fractions
Classifying Problems
The denominator in \(\frac{4 x-11}{x^3-9 x^2}\) has:
A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors
Calculus with Fractions
Classifying Problems
The denominator in \(\frac{z^2+2 z+3}{(z-6)\left(z^2+4\right)}\) has:
A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors
Calculus with Fractions
Classifying Problems
The denominator in \(\frac{z^2+2 z+3}{(z-6)\left(z^2+4\right)}\) has:
A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors
Calculus with Fractions
Classifying Problems
The denominator in \(\frac{x^3+10 x^2+3 x+36}{(x-1)\left(x^2+4\right)^2}\) has:
A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors
Calculus with Fractions
Classifying Problems
The denominator in \(\frac{x^3+10 x^2+3 x+36}{(x-1)\left(x^2+4\right)^2}\) has:
A. Non Repeating Linear Factors
B. Repeating Linear Factors
C. Non Repeating Irreducible Quadratic Factors
D. Repeating Irreducible Quadratic Factors
Calculus with Fractions
Classifying Problems
\(\frac{3 x+11}{(x-3)(x+2)}\) would decompose to
A. \(\frac{A}{x-3}+\frac{B}{x+2}\)
B. \(\frac{A}{x-3}+\frac{Bx}{x+2}\)
C. \(\frac{A}{x-3}+\frac{Bx+C}{x+2}\)
D. \(\frac{Ax+B}{x-3}+\frac{Cx+D}{x+2}\)
Calculus with Fractions
Classifying Problems
\(\frac{3 x+11}{(x-3)(x+2)}\) would decompose to
A. \(\frac{A}{x-3}+\frac{B}{x+2}\)
B. \(\frac{A}{x-3}+\frac{Bx}{x+2}\)
C. \(\frac{A}{x-3}+\frac{Bx+C}{x+2}\)
D. \(\frac{Ax+B}{x-3}+\frac{Cx+D}{x+2}\)
Calculus with Fractions
Classifying Problems
\(\frac{2+x^4}{x(x^2+9)}\) would decompose to
A. \(\frac{A}{x}+\frac{B}{x^2+9}\)
B. \(\frac{Ax+B}{x}+\frac{C}{x^2+9}\)
C. \(\frac{A}{x}+\frac{Bx+C}{x^2+9}\)
D. \(\frac{Ax+B}{x}+\frac{Cx+D}{x^2+9}\)
Calculus with Fractions
Classifying Problems
\(\frac{2+x^4}{x(x^2+9)}\) would decompose to
A. \(\frac{A}{x}+\frac{B}{x^2+9}\)
B. \(\frac{Ax+B}{x}+\frac{C}{x^2+9}\)
C. \(\frac{A}{x}+\frac{Bx+C}{x^2+9}\)
D. \(\frac{Ax+B}{x}+\frac{Cx+D}{x^2+9}\)
Calculus with Fractions
Classifying Problems
\(\frac{3 x+11}{(x+2)(x+2)}\) would decompose to
A. \(\frac{A}{x+2}+\frac{B}{x+2}\)
B. \(\frac{A}{x+2}+\frac{B}{(x+2)^2}\)
C. \(\frac{A}{x+2}+\frac{Bx+C}{x+2}\)
D. \(\frac{Ax+B}{x+2}+\frac{Cx+D}{(x+2)^2}\)
Calculus with Fractions
Classifying Problems
\(\frac{3 x+11}{(x+2)(x+2)}\) would decompose to
A. \(\frac{A}{x+2}+\frac{B}{x+2}\)
B. \(\frac{A}{x+2}+\frac{B}{(x+2)^2}\)
C. \(\frac{A}{x+2}+\frac{Bx+C}{x+2}\)
D. \(\frac{Ax+B}{x+2}+\frac{Cx+D}{(x+2)^2}\)
Calculus with Fractions
Classifying Problems
\(\frac{3 x+11}{x^2+6x+9}\) would decompose to
A. \(\frac{A}{x+3}+\frac{B}{x+3}\)
B. \(\frac{A}{x+3}+\frac{B}{(x+3)^2}\)
C. \(\frac{A}{x+3}+\frac{Bx+C}{x+3}\)
D. \(\frac{Ax+B}{x+3}+\frac{Cx+D}{(x+3)^2}\)
Calculus with Fractions
\(\frac{3 x+11}{x^2+6x+9}\) would decompose to
A. \(\frac{A}{x+3}+\frac{B}{x+3}\)
B. \(\frac{A}{x+3}+\frac{B}{(x+3)^2}\)
C. \(\frac{A}{x+3}+\frac{Bx+C}{x+3}\)
D. \(\frac{Ax+B}{x+3}+\frac{Cx+D}{(x+3)^2}\)
Classifying Problems
Calculus with Fractions
Classifying Problems
\(\frac{4 x-11}{x^3-9 x^2}\) would decompose to
A. \(\frac{A}{x^2}+\frac{B}{x-9}\)
B. \(\frac{Ax+B}{x^2}+\frac{C}{x-9}\)
C. \(\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x-9}\)
D. \(\frac{A}{x}+\frac{Bx+C}{x^2}+\frac{D}{x-9}\)
Calculus with Fractions
Classifying Problems
\(\frac{4 x-11}{x^3-9 x^2}\) would decompose to
A. \(\frac{A}{x^2}+\frac{B}{x-9}\)
B. \(\frac{Ax+B}{x^2}+\frac{C}{x-9}\)
C. \(\frac{A}{x}+\frac{B}{x^2}+\frac{C}{x-9}\)
D. \(\frac{A}{x}+\frac{Bx+C}{x^2}+\frac{D}{x-9}\)
Calculus with Fractions
Classifying Problems
\(\frac{z^2+2 z+3}{(z-6)\left(z^2+4\right)}\) would decompose to
A. \(\frac{A}{z-6}+\frac{B}{z^2+4}\)
B. \(\frac{A}{z-6}+\frac{Bx+C}{z^2+4}\)
C. \(\frac{Ax+B}{z-6}+\frac{Cx+D}{z+2}+\frac{Ex+F}{z-2}\)
D. \(\frac{A}{z-6}+\frac{B}{z+2}+\frac{C}{z-2}\)
Calculus with Fractions
\(\frac{z^2+2 z+3}{(z-6)\left(z^2+4\right)}\) would decompose to
A. \(\frac{A}{z-6}+\frac{B}{z^2+4}\)
B. \(\frac{A}{z-6}+\frac{Bx+C}{z^2+4}\)
C. \(\frac{Ax+B}{z-6}+\frac{Cx+D}{z+2}+\frac{Ex+F}{z-2}\)
D. \(\frac{A}{z-6}+\frac{B}{z+2}+\frac{C}{z-2}\)
Classifying Problems
Calculus with Fractions
Classifying Problems
\(\frac{x^3+10 x^2+3 x+36}{(x-1)\left(x^2+4\right)^2}\) would decompose to
A. \(\frac{A}{x-1}+\frac{B}{x^2+4}\)
B. \(\frac{A}{x-1}+\frac{Bx+C}{(x^2+4)^2}\)
C. \(\frac{A}{x-1}+\frac{Bx+C}{x^2+4}+\frac{Ex+F}{(x^2+4)^2}\)
D. \(\frac{A}{x-1}+\frac{B}{x^2+4}+\frac{C}{(x^2+4)^2}\)
Calculus with Fractions
Classifying Problems
\(\frac{x^3+10 x^2+3 x+36}{(x-1)\left(x^2+4\right)^2}\) would decompose to
A. \(\frac{A}{x-1}+\frac{B}{x^2+4}\)
B. \(\frac{A}{x-1}+\frac{Bx+C}{(x^2+4)^2}\)
C. \(\frac{A}{x-1}+\frac{Bx+C}{x^2+4}+\frac{Ex+F}{(x^2+4)^2}\)
D. \(\frac{A}{x-1}+\frac{B}{x^2+4}+\frac{C}{(x^2+4)^2}\)
Calculus with Fractions
In Class Problems
Group 1: \(\int \frac{5 x-7}{x^2-3 x+2} d x\)
Group 2: \(\int \frac{3}{x^3-3 x^2} d x\)
Group 3: \(\int \frac{x+1}{x^2-25} d x\)
Group 4: \(\int \frac{2}{x^3(x+1)} d x\)
Group 5: \(\int \frac{4}{(x+1)\left(x^2+1\right)} d x\)
Group 6: \(\int \frac{x^3}{x^2+8x+16} d x\)
\pi \cdot \pi
Calculus with Fractions
By Anurag Katyal
