Basic Integration Rules
| Rule | Formula | Example |
|---|---|---|
| Constant Rule | $\int k , dx = kx + C$ | $\int 5 , dx = 5x + C$ |
| Power Rule | $\int x^n , dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$) | $\int x^3 , dx = \frac{x^4}{4} + C$ |
| Logarithm Rule | $\int \frac{1}{x} , dx = \ln|x| + C$ | $\int \frac{1}{x} , dx = \ln|x| + C$ |
| Sum/Difference Rule | $\int [f(x) \pm g(x)] , dx = \int f(x) , dx \pm \int g(x) , dx$ | $\int (x^2 + \sin x) , dx = \frac{x^3}{3} – \cos x + C$ |
| Constant Multiple Rule | $\int kf(x) , dx = k\int f(x) , dx$ | $\int 3x^2 , dx = 3 \cdot \frac{x^3}{3} = x^3 + C$ |
Common Function Integrals
Basic Functions
| Function | Integral |
|---|---|
| $x^n$ (n ≠-1) | $\frac{x^{n+1}}{n+1} + C$ |
| $\frac{1}{x}$ | $\ln|x| + C$ |
| $\frac{1}{ax+b}$ | $\frac{1}{a}\ln|ax+b| + C$ |
| $\frac{1}{(ax+b)^n}$ (n > 1) | $\frac{-1}{a(n-1)(ax+b)^{n-1}} + C$ |
| $\sqrt{x}$ | $\frac{2x^{3/2}}{3} + C$ |
| $\frac{1}{\sqrt{x}}$ | $2\sqrt{x} + C$ |
Exponential and Logarithmic Functions
| Function | Integral |
|---|---|
| $e^x$ | $e^x + C$ |
| $a^x$ | $\frac{a^x}{\ln(a)} + C$ |
| $e^{ax}$ | $\frac{e^{ax}}{a} + C$ |
| $\ln(x)$ | $x\ln(x) – x + C$ |
| $x\ln(x)$ | $\frac{x^2\ln(x)}{2} – \frac{x^2}{4} + C$ |
| $\frac{1}{x\ln(x)}$ | $\ln|\ln(x)| + C$ |
Trigonometric Functions
| Function | Integral |
|---|---|
| $\sin(x)$ | $-\cos(x) + C$ |
| $\cos(x)$ | $\sin(x) + C$ |
| $\tan(x)$ | $-\ln|\cos(x)| + C$ or $\ln|\sec(x)| + C$ |
| $\cot(x)$ | $\ln|\sin(x)| + C$ |
| $\sec(x)$ | $\ln|\sec(x) + \tan(x)| + C$ |
| $\csc(x)$ | $\ln|\csc(x) – \cot(x)| + C$ |
| $\sec^2(x)$ | $\tan(x) + C$ |
| $\csc^2(x)$ | $-\cot(x) + C$ |
| $\sec(x)\tan(x)$ | $\sec(x) + C$ |
| $\csc(x)\cot(x)$ | $-\csc(x) + C$ |
Inverse Trigonometric Functions
| Function | Integral |
|---|---|
| $\frac{1}{\sqrt{1-x^2}}$ | $\arcsin(x) + C$ |
| $\frac{1}{1+x^2}$ | $\arctan(x) + C$ |
| $\frac{1}{\sqrt{x^2-1}}$ | $\ln|x + \sqrt{x^2-1}| + C$ or $\text{arcsec}(x) + C$ for $x > 1$ |
| $\frac{1}{x\sqrt{1-x^2}}$ | $-\text{arccsc}(x) + C$ |
| $\frac{1}{x\sqrt{x^2-1}}$ | $-\text{arcsec}(x) + C$ |
| $\frac{1}{\sqrt{a^2-x^2}}$ | $\arcsin\left(\frac{x}{a}\right) + C$ |
Special Integration Patterns
Integrals Involving $a^2 + x^2$
| Function | Integral |
|---|---|
| $\frac{1}{a^2+x^2}$ | $\frac{1}{a}\arctan\left(\frac{x}{a}\right) + C$ |
| $\frac{x}{a^2+x^2}$ | $\frac{1}{2}\ln(a^2+x^2) + C$ |
| $\frac{x^2}{a^2+x^2}$ | $x – a\arctan\left(\frac{x}{a}\right) + C$ |
| $\frac{1}{(a^2+x^2)^2}$ | $\frac{x}{2a^2(a^2+x^2)} + \frac{1}{2a^3}\arctan\left(\frac{x}{a}\right) + C$ |
| $\sqrt{a^2+x^2}$ | $\frac{x\sqrt{a^2+x^2}}{2} + \frac{a^2}{2}\ln\left |
| $\frac{1}{\sqrt{a^2+x^2}}$ | $\ln\left |
Integrals Involving $a^2 – x^2$
| Function | Integral |
|---|---|
| $\frac{1}{a^2-x^2}$ | $\frac{1}{2a}\ln\left |
| $\frac{x}{a^2-x^2}$ | $-\frac{1}{2}\ln|a^2-x^2| + C$ |
| $\frac{1}{(a^2-x^2)^2}$ | $\frac{x}{2a^2(a^2-x^2)} + \frac{1}{4a^3}\ln\left |
| $\sqrt{a^2-x^2}$ | $\frac{x\sqrt{a^2-x^2}}{2} + \frac{a^2}{2}\arcsin\left(\frac{x}{a}\right) + C$ |
| $\frac{1}{\sqrt{a^2-x^2}}$ | $\arcsin\left(\frac{x}{a}\right) + C$ |
| $x\sqrt{a^2-x^2}$ | $-\frac{(a^2-x^2)^{3/2}}{3} + C$ |
Integrals Involving $x^2 – a^2$
| Function | Integral |
|---|---|
| $\frac{1}{x^2-a^2}$ | $\frac{1}{2a}\ln\left |
| $\frac{1}{(x^2-a^2)^2}$ | $-\frac{x}{2a^2(x^2-a^2)} + \frac{1}{4a^3}\ln\left |
| $\sqrt{x^2-a^2}$ | $\frac{x\sqrt{x^2-a^2}}{2} – \frac{a^2}{2}\ln\left |
| $\frac{1}{\sqrt{x^2-a^2}}$ | $\ln\left |
| $x\sqrt{x^2-a^2}$ | $\frac{(x^2-a^2)^{3/2}}{3} + C$ |
Powers of Trigonometric Functions
| Function | Integral |
|---|---|
| $\sin^2(x)$ | $\frac{x}{2} – \frac{\sin(2x)}{4} + C$ |
| $\cos^2(x)$ | $\frac{x}{2} + \frac{\sin(2x)}{4} + C$ |
| $\tan^2(x)$ | $\tan(x) – x + C$ |
| $\sin^n(x)$ (n even) | Use $\sin^2(x) = \frac{1-\cos(2x)}{2}$ |
| $\sin^n(x)$ (n odd ≥ 3) | Use $\sin^n(x) = \sin^{n-2}(x)(1-\cos^2(x))$ |
| $\cos^n(x)$ (n even) | Use $\cos^2(x) = \frac{1+\cos(2x)}{2}$ |
| $\cos^n(x)$ (n odd ≥ 3) | Use $\cos^n(x) = \cos^{n-2}(x)(1-\sin^2(x))$ |
| $\sin(x)\cos(x)$ | $-\frac{\cos(2x)}{4} + C$ or $\frac{\sin^2(x)}{2} + C$ |
Products of Trigonometric Functions
| Function | Integral |
|---|---|
| $\sin(mx)\sin(nx)$ | $\frac{\sin((m-n)x)}{2(m-n)} – \frac{\sin((m+n)x)}{2(m+n)} + C$ (for m ≠n, m ≠-n) |
| $\cos(mx)\cos(nx)$ | $\frac{\sin((m-n)x)}{2(m-n)} + \frac{\sin((m+n)x)}{2(m+n)} + C$ (for m ≠n, m ≠-n) |
| $\sin(mx)\cos(nx)$ | $-\frac{\cos((m-n)x)}{2(m-n)} – \frac{\cos((m+n)x)}{2(m+n)} + C$ (for m ≠n, m ≠-n) |
| $\sin(x)\sin(nx)$ | $\frac{\cos((n-1)x)}{2(n-1)} – \frac{\cos((n+1)x)}{2(n+1)} + C$ (for n ≠1, n ≠-1) |
| $\cos(x)\cos(nx)$ | $\frac{\cos((n-1)x)}{2(n-1)} + \frac{\cos((n+1)x)}{2(n+1)} + C$ (for n ≠1, n ≠-1) |
| $\sin(x)\cos(nx)$ | $\frac{\sin((n-1)x)}{2(n-1)} + \frac{\sin((n+1)x)}{2(n+1)} + C$ (for n ≠1, n ≠-1) |
Integration Techniques
Substitution Method (U-Substitution)
For integrals of the form $\int f(g(x))g'(x) , dx$:
- Let u = g(x)
- Find du = g'(x) dx
- Substitute to get $\int f(u) , du$
- Integrate with respect to u
- Substitute back to get answer in terms of x
Example: $\int 2x\cos(x^2) , dx$
- Let u = x²
- du = 2x dx
- $\int 2x\cos(x^2) , dx = \int \cos(u) , du = \sin(u) + C = \sin(x^2) + C$
Integration by Parts
For integrals of the form $\int u(x)v'(x) , dx$:
Formula: $\int u(x)v'(x) , dx = u(x)v(x) – \int v(x)u'(x) , dx$
LIATE method for choosing u (in order of preference):
- L: Logarithmic functions
- I: Inverse trigonometric functions
- A: Algebraic functions
- T: Trigonometric functions
- E: Exponential functions
Example: $\int x\sin(x) , dx$
- u = x, dv = sin(x) dx
- du = dx, v = -cos(x)
- $\int x\sin(x) , dx = -x\cos(x) – \int (-\cos(x)) , dx = -x\cos(x) + \int \cos(x) , dx$
- $= -x\cos(x) + \sin(x) + C$
Trigonometric Substitution
Used for integrals involving:
| Expression | Substitution | Identity |
|---|---|---|
| $\sqrt{a^2-x^2}$ | x = a sin(θ) | $\sqrt{a^2-x^2} = a\cos(θ)$ |
| $\sqrt{x^2-a^2}$ | x = a sec(θ) | $\sqrt{x^2-a^2} = a\tan(θ)$ |
| $\sqrt{x^2+a^2}$ | x = a tan(θ) | $\sqrt{x^2+a^2} = a\sec(θ)$ |
Example: $\int \frac{1}{\sqrt{9-x^2}} , dx$
- Let x = 3sin(θ), dx = 3cos(θ) dθ
- $\sqrt{9-x^2} = \sqrt{9-9\sin^2(θ)} = \sqrt{9\cos^2(θ)} = 3\cos(θ)$
- $\int \frac{1}{\sqrt{9-x^2}} , dx = \int \frac{3\cos(θ)}{3\cos(θ)} , dθ = \int dθ = θ + C = \arcsin\left(\frac{x}{3}\right) + C$
Common Integration Patterns
Rational Functions
| Function | Integral |
|---|---|
| $\frac{1}{(x-a)(x-b)}$ | $\frac{1}{a-b}\ln\left|\frac{x-a}{x-b}\right| + C$ |
| $\frac{1}{x(x-a)}$ | $\frac{1}{a}\ln\left|\frac{x}{x-a}\right| + C$ |
| $\frac{1}{x^2-a^2}$ | $\frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right| + C$ |
| $\frac{1}{x^2+a^2}$ | $\frac{1}{a}\arctan\left(\frac{x}{a}\right) + C$ |
Completing the Square
For integrals involving quadratic expressions:
- Complete the square in the denominator
- Use substitution to simplify the integral
- Apply standard integration formulas
Example: $\int \frac{1}{x^2+6x+13} , dx$
- Complete the square: $x^2+6x+13 = (x+3)^2 + 4$
- Let u = x+3, then du = dx
- $\int \frac{1}{x^2+6x+13} , dx = \int \frac{1}{(u)^2+4} , du = \frac{1}{2}\arctan\left(\frac{u}{2}\right) + C = \frac{1}{2}\arctan\left(\frac{x+3}{2}\right) + C$
Tips for Successful Integration
- Look for patterns that match standard forms
- Try substitution first for composite functions where one function’s derivative appears
- Use integration by parts for products where one factor is easier to differentiate than integrate
- For powers of trig functions:
- For even powers, use double-angle formulas
- For odd powers, separate one factor and use substitution
- For rational functions, use partial fractions decomposition
- For expressions with square roots, consider trigonometric substitution
- For improper integrals, use limits to handle infinite bounds or singularities
Integration Tables Reference
Key Formulas to Remember
| Function | Integral |
|---|---|
| $\frac{1}{\sqrt{1-x^2}}$ | $\arcsin(x) + C$ |
| $\frac{1}{1+x^2}$ | $\arctan(x) + C$ |
| $\frac{1}{\sqrt{x^2-1}}$ | $\ln|x + \sqrt{x^2-1}| + C$ |
| $e^{ax}$ | $\frac{1}{a}e^{ax} + C$ |
| $\sin(ax)$ | $-\frac{1}{a}\cos(ax) + C$ |
| $\cos(ax)$ | $\frac{1}{a}\sin(ax) + C$ |
| $\ln(x)$ | $x\ln(x) – x + C$ |
| $\sec^2(x)$ | $\tan(x) + C$ |
| $\csc^2(x)$ | $-\cot(x) + C$ |
| $\sec(x)\tan(x)$ | $\sec(x) + C$ |
| $\csc(x)\cot(x)$ | $-\csc(x) + C$ |
| $\frac{1}{x\ln(x)}$ | $\ln|\ln(x)| + C$ |
Applications of Integration
Area Calculation
- Area under curve: $A = \int_a^b f(x) , dx$ (for f(x) ≥ 0)
- Area between curves: $A = \int_a^b [f(x) – g(x)] , dx$ (for f(x) ≥ g(x))
Volume Calculation
- Disk method (around x-axis): $V = \pi \int_a^b [f(x)]^2 , dx$
- Washer method (around x-axis): $V = \pi \int_a^b [(f(x))^2 – (g(x))^2] , dx$
- Shell method (around y-axis): $V = 2\pi \int_a^b x \cdot f(x) , dx$
Arc Length and Surface Area
- Arc length: $L = \int_a^b \sqrt{1 + [f'(x)]^2} , dx$
- Surface area of revolution (around x-axis): $S = 2\pi \int_a^b f(x) \sqrt{1 + [f'(x)]^2} , dx$
Definite Integral Properties
- $\int_a^b f(x) , dx = -\int_b^a f(x) , dx$
- $\int_a^b f(x) , dx = \int_a^c f(x) , dx + \int_c^b f(x) , dx$
- If f(x) is even: $\int_{-a}^a f(x) , dx = 2\int_0^a f(x) , dx$
- If f(x) is odd: $\int_{-a}^a f(x) , dx = 0$
Improper Integrals
Type 1 (Infinite bounds):
- $\int_a^{\infty} f(x) , dx = \lim_{t \to \infty} \int_a^t f(x) , dx$
- $\int_{-\infty}^b f(x) , dx = \lim_{t \to -\infty} \int_t^b f(x) , dx$
Type 2 (Discontinuous integrands):
- $\int_a^b f(x) , dx = \lim_{t \to c^-} \int_a^t f(x) , dx + \lim_{t \to c^+} \int_t^b f(x) , dx$ (Where f(x) has a discontinuity at x = c, with a < c < b)
This comprehensive cheatsheet covers the twelve most essential integral patterns and techniques in calculus. Use it as a quick reference during problem-solving to identify the appropriate method and formula for evaluating various types of integrals.