Derivatives and Integration Formulas
1. Basic Derivative Formulas
Constant and Power Functions
- \(\frac{d}{dx}(c) = 0\) where \(c\) is constant
- \(\frac{d}{dx}(x) = 1\)
- \(\frac{d}{dx}(x^n) = nx^{n-1}\)
- \(\frac{d}{dx}(\sqrt{x}) = \frac{1}{2\sqrt{x}}\)
- \(\frac{d}{dx}\left(\frac{1}{x}\right) = -\frac{1}{x^2}\)
- \(\frac{d}{dx}(x^{-n}) = -nx^{-n-1}\)
Exponential and Logarithmic Functions
- \(\frac{d}{dx}(e^x) = e^x\)
- \(\frac{d}{dx}(a^x) = a^x \ln a\)
- \(\frac{d}{dx}(\ln x) = \frac{1}{x}\)
- \(\frac{d}{dx}(\log_a x) = \frac{1}{x \ln a}\)
- \(\frac{d}{dx}(e^{f(x)}) = f'(x) \cdot e^{f(x)}\)
- \(\frac{d}{dx}(\ln(f(x))) = \frac{f'(x)}{f(x)}\)
Trigonometric Functions
- \(\frac{d}{dx}(\sin x) = \cos x\)
- \(\frac{d}{dx}(\cos x) = -\sin x\)
- \(\frac{d}{dx}(\tan x) = \sec^2 x\)
- \(\frac{d}{dx}(\cot x) = -\csc^2 x\)
- \(\frac{d}{dx}(\sec x) = \sec x \tan x\)
- \(\frac{d}{dx}(\csc x) = -\csc x \cot x\)
Inverse Trigonometric Functions
- \(\frac{d}{dx}(\sin^{-1} x) = \frac{1}{\sqrt{1-x^2}}\)
- \(\frac{d}{dx}(\cos^{-1} x) = \frac{-1}{\sqrt{1-x^2}}\)
- \(\frac{d}{dx}(\tan^{-1} x) = \frac{1}{1+x^2}\)
- \(\frac{d}{dx}(\cot^{-1} x) = \frac{-1}{1+x^2}\)
- \(\frac{d}{dx}(\sec^{-1} x) = \frac{1}{|x|\sqrt{x^2-1}}\)
- \(\frac{d}{dx}(\csc^{-1} x) = \frac{-1}{|x|\sqrt{x^2-1}}\)
2. Differentiation Rules
Basic Rules
Sum/Difference Rule:
\[\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)\]Constant Multiple Rule:
\[\frac{d}{dx}[cf(x)] = c \cdot f'(x)\]Product Rule:
\[\frac{d}{dx}[f(x) \cdot g(x)] = f'(x) \cdot g(x) + f(x) \cdot g'(x)\]Quotient Rule:
\[\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{f'(x) \cdot g(x) - f(x) \cdot g'(x)}{[g(x)]^2}\]Chain Rule:
\[\frac{d}{dx}[f(g(x))] = f'(g(x)) \cdot g'(x)\]3. Basic Integration Formulas
Power Functions
- \(\int k \, dx = kx + C\)
- \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\) (where \(n \neq -1\))
- \(\int \frac{1}{x} \, dx = \ln|x| + C\)
- \(\int \sqrt{x} \, dx = \frac{2}{3}x^{3/2} + C\)
- \(\int \frac{1}{\sqrt{x}} \, dx = 2\sqrt{x} + C\)
- \(\int \frac{1}{x^2} \, dx = -\frac{1}{x} + C\)
Exponential and Logarithmic Functions
- \(\int e^x \, dx = e^x + C\)
- \(\int a^x \, dx = \frac{a^x}{\ln a} + C\)
- \(\int e^{ax} \, dx = \frac{e^{ax}}{a} + C\)
- \(\int \frac{1}{x} \, dx = \ln|x| + C\)
- \(\int \ln x \, dx = x\ln x - x + C\)
- \(\int \frac{1}{ax+b} \, dx = \frac{1}{a}\ln|ax+b| + C\)
Trigonometric Functions
- \(\int \sin x \, dx = -\cos x + C\)
- \(\int \cos x \, dx = \sin x + C\)
- \(\int \tan x \, dx = -\ln|\cos x| + C\)
- \(\int \cot x \, dx = \ln|\sin x| + C\)
- \(\int \sec x \, dx = \ln|\sec x + \tan x| + C\)
- \(\int \csc x \, dx = -\ln|\csc x + \cot x| + C\)
- \(\int \sec^2 x \, dx = \tan x + C\)
- \(\int \csc^2 x \, dx = -\cot x + C\)
- \(\int \sec x \tan x \, dx = \sec x + C\)
- \(\int \csc x \cot x \, dx = -\csc x + C\)
Inverse Trigonometric Functions
- \(\int \frac{1}{\sqrt{1-x^2}} \, dx = \sin^{-1} x + C\)
- \(\int \frac{-1}{\sqrt{1-x^2}} \, dx = \cos^{-1} x + C\)
- \(\int \frac{1}{1+x^2} \, dx = \tan^{-1} x + C\)
- \(\int \frac{-1}{1+x^2} \, dx = \cot^{-1} x + C\)
- \(\int \frac{1}{x\sqrt{x^2-1}} \, dx = \sec^{-1} x + C\)
- \(\int \frac{-1}{x\sqrt{x^2-1}} \, dx = \csc^{-1} x + C\)
4. Integration Rules
Basic Integration Rules
Sum/Difference Rule:
\[\int [f(x) \pm g(x)] \, dx = \int f(x) \, dx \pm \int g(x) \, dx\]Constant Multiple Rule:
\[\int k \cdot f(x) \, dx = k \int f(x) \, dx\]Integration by Parts:
\[\int u \, dv = uv - \int v \, du\]Substitution Rule:
\[\int f(g(x)) \cdot g'(x) \, dx = \int f(u) \, du\] where \(u = g(x)\)5. Standard Integral Forms
Rational Functions
- \(\int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a}\tan^{-1}\left(\frac{x}{a}\right) + C\)
- \(\int \frac{1}{x^2 - a^2} \, dx = \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right| + C\)
- \(\int \frac{1}{a^2 - x^2} \, dx = \frac{1}{2a}\ln\left|\frac{a+x}{a-x}\right| + C\)
- \(\int \frac{x}{x^2 + a^2} \, dx = \frac{1}{2}\ln(x^2 + a^2) + C\)
Square Root Functions
- \(\int \frac{1}{\sqrt{x^2 + a^2}} \, dx = \ln|x + \sqrt{x^2 + a^2}| + C\)
- \(\int \frac{1}{\sqrt{x^2 - a^2}} \, dx = \ln|x + \sqrt{x^2 - a^2}| + C\)
- \(\int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \sin^{-1}\left(\frac{x}{a}\right) + C\)
- \(\int \sqrt{a^2 - x^2} \, dx = \frac{x}{2}\sqrt{a^2-x^2} + \frac{a^2}{2}\sin^{-1}\left(\frac{x}{a}\right) + C\)
6. Trigonometric Integrals
Powers of Sine and Cosine
- \(\int \sin^2 x \, dx = \frac{x}{2} - \frac{\sin 2x}{4} + C\)
- \(\int \cos^2 x \, dx = \frac{x}{2} + \frac{\sin 2x}{4} + C\)
- \(\int \sin^n x \cos x \, dx = \frac{\sin^{n+1} x}{n+1} + C\)
- \(\int \cos^n x \sin x \, dx = -\frac{\cos^{n+1} x}{n+1} + C\)
Product Formulas
- \(\int \sin mx \cos nx \, dx = -\frac{\cos(m+n)x}{2(m+n)} - \frac{\cos(m-n)x}{2(m-n)} + C\)
- \(\int \sin mx \sin nx \, dx = \frac{\sin(m-n)x}{2(m-n)} - \frac{\sin(m+n)x}{2(m+n)} + C\)
- \(\int \cos mx \cos nx \, dx = \frac{\sin(m+n)x}{2(m+n)} + \frac{\sin(m-n)x}{2(m-n)} + C\)
7. Integration Techniques
Integration by Parts - LIATE Rule
Choose \(u\) in the order: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential
Partial Fractions
For \(\frac{P(x)}{Q(x)}\) where degree of \(P(x) < \) degree of \(Q(x)\):
- Linear factors: \(\frac{A}{x-a} + \frac{B}{x-b}\)
- Repeated linear: \(\frac{A}{x-a} + \frac{B}{(x-a)^2}\)
- Quadratic: \(\frac{Ax+B}{x^2+bx+c}\)
Trigonometric Substitution
- For \(\sqrt{a^2 - x^2}\): use \(x = a\sin\theta\)
- For \(\sqrt{a^2 + x^2}\): use \(x = a\tan\theta\)
- For \(\sqrt{x^2 - a^2}\): use \(x = a\sec\theta\)
8. Definite Integration
Fundamental Theorem of Calculus
\[\int_a^b f(x) \, dx = F(b) - F(a)\] where \(F'(x) = f(x)\)Properties of Definite Integrals
- \(\int_a^a f(x) \, dx = 0\)
- \(\int_a^b f(x) \, dx = -\int_b^a f(x) \, dx\)
- \(\int_a^b [f(x) + g(x)] \, dx = \int_a^b f(x) \, dx + \int_a^b g(x) \, dx\)
- \(\int_a^b k \cdot f(x) \, dx = k \int_a^b f(x) \, dx\)
- \(\int_a^c f(x) \, dx = \int_a^b f(x) \, dx + \int_b^c f(x) \, dx\)
Special Properties
- \(\int_0^a f(x) \, dx = \int_0^a f(a-x) \, dx\)
- \(\int_{-a}^a f(x) \, dx = 2\int_0^a f(x) \, dx\) if \(f(x)\) is even
- \(\int_{-a}^a f(x) \, dx = 0\) if \(f(x)\) is odd
- \(\int_0^{2a} f(x) \, dx = \int_0^a [f(x) + f(2a-x)] \, dx\)
9. Applications
Area Under Curve
Area = \(\int_a^b f(x) \, dx\)
Area between curves:
\(\int_a^b [f(x) - g(x)] \, dx\)
Volume of Revolution
Disk method:
\(V = \pi \int_a^b [f(x)]^2 \, dx\)
Washer method:
\(V = \pi \int_a^b [R(x)]^2 - [r(x)]^2 \, dx\)
Remember: Always add the constant of integration \(+C\) for indefinite integrals!