List of Integral Formulas

(i) Basic Integral Formulas

\[\begin{align} & (1)\int [f(x) \pm g(x)] \, dx = \int f(x)\, dx \pm \int g(x)\, dx \\ & (2)\int a f(x)\, dx = a \int f(x)\, dx + C, \text{where a is a constant} \\ & (3)\int dx = x + C\\ & (4)\int a\, dx = ax + C\\ & (5)\int x^n dx = \frac{x^{n+1}}{n+1} + C,\; n \neq -1\\ & (6)\int x^{-1} dx = \int \frac{1}{x} dx = \ln|x| + C\\ & (7)\int \sin x \, dx = -\cos x + C\\ & (8)\int \cos x \, dx = \sin x + C\\ & (9)\int \sec^2x \, dx = \tan x + C\\ & (10)\int \sec x \tan x \, dx = \sec x + C\\ & (11)\int \csc^2x \, dx = -\cot x + C\\ & (12)\int \csc x \cot x \, dx = -\csc x + C\\ & (13)\int e^x dx = e^x + C\\ & (14)\int a^x dx = \frac{a^x}{\ln a} + C,\; a>0\\ & (15)\int \frac{dx}{\sqrt{1-x^2}} = \sin^{-1}x + C\\ & (16)\int \frac{dx}{\sqrt{1-x^2}} = -\cos^{-1}x + C\\ & (17)\int \frac{dx}{1+x^2} = \tan^{-1}x + C\\ & (18)\int \frac{dx}{1+x^2} = -\cot^{-1}x + C\\ & (19)\int \frac{dx}{x\sqrt{x^2-1}} = \sec^{-1}x + C\\ & (20)\int \frac{dx}{x\sqrt{x^2-1}} = -\csc^{-1}x + C\\ & (21)\int \tan x dx = \ln|\sec x| + C = -\ln|\cos x| + C\\ & (22)\int \cot x dx = \ln|\sin x| + C\\ & (23)\int \sec x dx = \ln|\sec x + \tan x| + C\\ & (24)\int \csc x dx = \ln|\csc x - \cot x| + C = \ln|\tan(x/2)| + C \end{align}\]

(ii) Integral of Some Special Functions

\[\begin{align} & (1)\int \frac{dx}{x^2-a^2} = \frac{1}{2a}\ln\Big|\frac{x-a}{x+a}\Big| + C \\ & (2)\int \frac{dx}{a^2-x^2} = \frac{1}{2a}\ln\Big|\frac{a+x}{a-x}\Big| + C \\ & (3)\int \frac{dx}{x^2+a^2} = \frac{1}{a}\tan^{-1}\Big(\frac{x}{a}\Big) + C \\ & (4)\int \frac{dx}{\sqrt{x^2+a^2}} = \ln|x+\sqrt{x^2+a^2}| + C \\ & (5)\int \frac{dx}{\sqrt{x^2-a^2}} = \ln|x+\sqrt{x^2-a^2}| + C \\ & (6)\int \frac{dx}{\sqrt{a^2-x^2}} = \sin^{-1}(x/a) + C \end{align}\]

(iii) Integrals by Partial Fraction

\[\begin{align} & (1)\frac{px+q}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b} \\ & (2)\frac{px+q}{(x-a)^2} = \frac{A}{x-a} + \frac{B}{(x-a)^2} \\ & (3)\frac{px^2+qx+r}{(x-a)(x-b)(x-c)} = \frac{A}{x-a} + \frac{B}{x-b} + \frac{C}{x-c} \\ & (4)\frac{px^2+qx+r}{(x-a)^2(x-b)} = \frac{A}{x-a} + \frac{B}{(x-a)^2} + \frac{C}{x-b} \\ & (5)\frac{px^2+qx+r}{(x-a)(x^2+bx+c)} = \frac{A}{x-a} + \frac{Bx+C}{x^2+bx+c} \end{align}\]

Note:
Rational Function = P(x)/Q(x), where Q(x) ≠ 0.
Proper Rational Function = deg(P) < deg(Q).
Improper Rational Function = deg(P) ≥ deg(Q).

Example: Convert improper to proper function:
$$\frac{5x^3-4x^2+3x-4}{x^2-2} = 5x-4 + \frac{21x-4}{x^2-2}$$

(iv) Integration by Parts

$$(1)\int u v dx = u\int v dx - \int \frac{du}{dx}\Big(\int v dx\Big) dx$$

Order of choosing first function (ILATE Rule):
I = Inverse Trig, L = Logarithm, A = Algebraic, T = Trig, E = Exponential

$$(2)\int e^x(f(x)+f'(x)) dx = e^x f(x) + C$$

(v) Other Special Integrals

\[\begin{align} & (1)\int \sqrt{x^2-a^2}\, dx = \frac{x}{2}\sqrt{x^2-a^2} - \frac{a^2}{2}\ln|x+\sqrt{x^2-a^2}| + C \\ & (2)\int \sqrt{x^2+a^2}\, dx = \frac{x}{2}\sqrt{x^2+a^2} + \frac{a^2}{2}\ln|x+\sqrt{x^2+a^2}| + C \\ & (3)\int \sqrt{a^2-x^2}\, dx = \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}(x/a) + C \\ & (4)\int e^{ax}\cos(bx) dx = \frac{e^{ax}}{a^2+b^2}(a\cos bx + b\sin bx) + C \\ & (5)\int e^{ax}\sin(bx) dx = \frac{e^{ax}}{a^2+b^2}(a\sin bx - b\cos bx) + C \end{align}\]

More Special Cases:
Integrals of the type $\int \frac{dx}{ax^2+bx+c}$, $\int \frac{dx}{\sqrt{ax^2+bx+c}}$, and $\int \frac{px+q}{ax^2+bx+c} dx$ can be solved by reducing to standard forms using substitution and coefficient comparison.