Partial Fractions
Partial Fractions Method for Indefinite Integrals:
 Principle: Used for decomposing a complex rational function into simpler fractions.
 Applicability: Typically applied when integrating rational functions (ratio of polynomials).
 Steps:
Steps for Partial Fractions Decomposition:

Proper Fraction:
 Ensure that the degree of the numerator is less than the degree of the denominator.

Factorization:
 Factorize the denominator into irreducible factors.

Partial Fractions Decomposition:
 Express the given rational function as a sum of partial fractions.
 If the factors are linear and distinct: $\frac{P(x)}{Q(x)}=\frac{A}{x{r}_{1}}+\frac{B}{x{r}_{2}}+\cdots +\frac{N}{x{r}_{n}}$
 If there are repeated factors: $\frac{P(x)}{Q(x)}=\frac{{A}_{1}}{x{r}_{1}}+\frac{{A}_{2}}{(x{r}_{1}{)}^{2}}+\cdots +\frac{{A}_{k}}{(x{r}_{1}{)}^{k}}$

Determining Coefficients:
 Find the unknown coefficients by various methods like equating coefficients, coverup method, or simultaneous equations.

Integration:
 Integrate each partial fraction term individually.
Example:
Let's solve the indefinite integral ∫ (4x + 7) / (${x}^{2}$+ 3x + 2) dx using the partial fractions method:
Steps:

Proper Fraction Check:
 The degree of the numerator (1) is less than the degree of the denominator (2). It's a proper fraction.

Factorization:
 Factorize the denominator: ${x}^{2}+3x+2=(x+1)(x+2)$.

Partial Fractions Decomposition:
 Express $(4x+7)\mathrm{/}({x}^{2}+3x+2)$ as:
$\frac{4x+7}{{x}^{2}+3x+2}=\frac{A}{x+1}+\frac{B}{x+2}$

Determining Coefficients:
 Multiply both sides by the denominator to solve for A and B.
 $4x+7=A(x+2)+B(x+1)$
 Substitute values of x to solve for A and B. (e.g., x = 1 and x = 2)
Solving for A and B gives: A = 3 and B = 1.

Integration:
 Rewrite the integral in terms of the partial fractions:
$\int \frac{4x+7}{{x}^{2}+3x+2}dx=\int \frac{3}{x+1}+\frac{1}{x+2}dx$
 Integrate each term individually:
$=3\cdot \mathrm{ln}\mathrm{\mid}x+1\mathrm{\mid}+\mathrm{ln}\mathrm{\mid}x+2\mathrm{\mid}+C$
Tips for Partial Fractions:
 Factorization is Key: Properly factorize the denominator before decomposing into partial fractions.
 Systematic Approach: Write down the partial fraction form and solve for unknown coefficients methodically.
 Check your Work: Ensure the decomposition is correct by recombining the partial fractions.