Calculus Unit 6: Partial Integration and Taylor Series

Fundamental Concept: Integration by parts is a technique derived from the product rule of differentiation, used to integrate the product of two functions. It effectively reverses the product rule.
Derivation and Formula: Given two differentiable functions $u = f(x)$ and $v = g(x)$ , the product rule states that $\frac{d}{dx}(uv) = u\frac{dv}{dx} + v\frac{du}{dx}$ . Integrating both sides with respect to $x$ gives $uv = \int u\,dv + \int v\,du$ . Rearranging this yields the formula: - $\int u\,dv = uv - \int v\,du$
Selection of $u$ (LIATE Rule): To simplify the resulting integral $\int v\,du$ , the choice of $u$ usually follows the LIATE priority scheme: - L: Logarithmic functions (e.g., $\ln(x)$ , $\log_2(x)$ ) - I: Inverse trigonometric functions (e.g., $\arctan(x)$ , $\arcsin(x)$ ) - A: Algebraic functions (e.g., $x^2$ , $3x+1$ , $\sqrt{x}$ , polynomial terms) - T: Trigonometric functions (e.g., $\sin(x)$ , $\cos(x)$ , $\sec^2(x)$ ) - E: Exponential functions (e.g., $e^x$ , $2^x$ )
Definite Integrals by Parts: For integration over the interval $[a, b]$ , the formula is applied as: - $\int_a^b u\,dv = [u(x)v(x)]_a^b - \int_a^b v\,du$
Repeated Integration and the Tabular Method: When the algebraic part is a high-degree polynomial (e.g., $x^3$ ), the tabular method is used to keep track of repeated steps. One column lists the derivatives of $u$ (ending at $0$ ) and the other lists the integrals of $dv$ , with alternating signs applied to the products.

Definition: This is an algebraic procedure used to decompose a rational function into a sum of simpler fractions, which can then be integrated individually. It is primarily used for proper rational functions where \text{deg}(P(x)) < \text{deg}(Q(x)).
Pre-requisite (Improper Fractions): If the degree of the numerator is greater than or equal to the degree of the denominator, polynomial long division must be performed first to obtain a polynomial plus a proper rational function.
Decomposition Cases: - Case 1: Distinct Linear Factors: If $Q(x)$ factors into $(x-r_1)(x-r_2)$ , then $\frac{P(x)}{Q(x)} = \frac{A}{x-r_1} + \frac{B}{x-r_2}$ . - Case 2: Repeated Linear Factors: If $Q(x)$ has a factor $(x-r)^k$ , the decomposition must include terms $\frac{A_1}{x-r} + \frac{A_2}{(x-r)^2} + \dots + \frac{A_k}{(x-r)^k}$ . - Case 3: Irreducible Quadratic Factors: If $Q(x)$ has a factor $(ax^2+bx+c)$ , the numerator is linear, of the form $Ax+B$ . - Case 4: Repeated Irreducible Quadratic Factors: For $(ax^2+bx+c)^k$ , the numerators take the form $\frac{A_i x + B_i}{(ax^2+bx+c)^i}$ .
Integration of Resulting Terms: Resulting fractions typically integrate into logarithmic functions (e.g., $\int \frac{1}{x-a}\,dx = \ln|x-a| + C$ ) or inverse trigonometric functions (e.g., $\int \frac{1}{x^2+1}\,dx = \arctan(x) + C$ ).

Mechanism: Partial integration involves integrating a function of several variables with respect to one variable while holding the other variables constant.
Indefinite Partial Integral: If $f(x, y)$ is integrated with respect to $x$ , the constant of integration is a function of $y$ : - $\int f(x, y)\,dx = F(x, y) + g(y)$ - Here, $\frac{\partial}{\partial x}[F(x, y) + g(y)] = f(x, y)$ .

Taylor Series Definition: A Taylor series is a representation of a function as an infinite sum of terms calculated from the values of its derivatives at a specific point $a$ . The series is given by: - $f(x) = f(a) + f'(a)(x-a) + \frac{f''(a)}{2!}(x-a)^2 + \frac{f'''(a)}{3!}(x-a)^3 + \dots$ - Sigma Notation: $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$
Maclaurin Series: This is the specific case where the series is centered at $a = 0$ : - $f(x) = \sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$
Taylor's Inequality and Error Estimation: The difference between the actual function and the $n$ -th degree Taylor polynomial $T_n(x)$ is the remainder $R_n(x)$ . According to Taylor’s Theorem, if $|f^{(n+1)}(x)| \le M$ for all $x$ in an interval, then: - $|R_n(x)| \le \frac{M}{(n+1)!}|x-a|^{n+1}$

Radius of Convergence ( $R$ ): The value such that the series converges if |x-a| < R and diverges if |x-a| > R.
Interval of Convergence: The set of all values for which the series converges, including potential endpoints where the series might converge or diverge.
Ratio Test for Convergence: To find the radius of convergence, examine the limit $L = \lim_{n \rightarrow \infty} |\frac{c_{n+1}(x-a)^{n+1}}{c_n(x-a)^n}| = |x-a| \lim_{n \rightarrow \infty} |\frac{c_{n+1}}{c_n}|$ . For convergence, L < 1.

Exponential Function: $e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \dots = \sum_{n=0}^{\infty} \frac{x^n}{n!}$ , with $R = \infty$ .
Sine Function: $\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \dots = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n+1}}{(2n+1)!}$ , with $R = \infty$ .
Cosine Function: $\cos(x) = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \dots = \sum_{n=0}^{\infty} (-1)^n \frac{x^{2n}}{(2n)!}$ , with $R = \infty$ .
Natural Logarithm: $\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \dots = \sum_{n=1}^{\infty} (-1)^{n-1} \frac{x^n}{n}$ , for -1 < x \le 1.
Geometric Series: $\frac{1}{1-x} = 1 + x + x^2 + x^3 + \dots = \sum_{n=0}^{\infty} x^n$ , for |x| < 1.