Integration is the reverse process of differentiation. It’s used to find areas, volumes, and solve differential equations.
Overview
graph TD
A[Integral Calculus] --> B[Indefinite Integrals]
A --> C[Definite Integrals]
B --> B1[Standard Forms]
B --> B2[Methods]
B2 --> B2a[Substitution]
B2 --> B2b[By Parts]
B2 --> B2c[Partial Fractions]
C --> C1[Properties]
C --> C2[Area Under Curves]Indefinite Integrals
Definition
$$\int f(x)\,dx = F(x) + C$$where $F'(x) = f(x)$ and $C$ is the constant of integration.
Standard Integrals
Basic Forms
| Function | Integral |
|---|---|
| $x^n$ (n ≠ -1) | $\frac{x^{n+1}}{n+1} + C$ |
| $\frac{1}{x}$ | $\ln |
| $e^x$ | $e^x + C$ |
| $a^x$ | $\frac{a^x}{\ln a} + C$ |
Trigonometric Integrals
| Function | Integral |
|---|---|
| $\sin x$ | $-\cos x + C$ |
| $\cos x$ | $\sin x + C$ |
| $\sec^2 x$ | $\tan x + C$ |
| $\csc^2 x$ | $-\cot x + C$ |
| $\sec x \tan x$ | $\sec x + C$ |
| $\csc x \cot x$ | $-\csc x + C$ |
| $\tan x$ | $\ln |
| $\cot x$ | $\ln |
| $\sec x$ | $\ln |
| $\csc x$ | $\ln |
Inverse Trigonometric Forms
$$\int \frac{1}{\sqrt{1-x^2}}\,dx = \sin^{-1}x + C$$ $$\int \frac{1}{1+x^2}\,dx = \tan^{-1}x + C$$ $$\int \frac{1}{x\sqrt{x^2-1}}\,dx = \sec^{-1}x + C$$Special Integrals
Type 1: $\int \frac{dx}{x^2 + a^2}$
$$\boxed{\int \frac{dx}{x^2 + a^2} = \frac{1}{a}\tan^{-1}\frac{x}{a} + C}$$Type 2: $\int \frac{dx}{x^2 - a^2}$
$$\boxed{\int \frac{dx}{x^2 - a^2} = \frac{1}{2a}\ln\left|\frac{x-a}{x+a}\right| + C}$$Type 3: $\int \frac{dx}{\sqrt{a^2 - x^2}}$
$$\boxed{\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\frac{x}{a} + C}$$Type 4: $\int \frac{dx}{\sqrt{x^2 + a^2}}$
$$\boxed{\int \frac{dx}{\sqrt{x^2 + a^2}} = \ln|x + \sqrt{x^2 + a^2}| + C}$$Type 5: $\int \frac{dx}{\sqrt{x^2 - a^2}}$
$$\boxed{\int \frac{dx}{\sqrt{x^2 - a^2}} = \ln|x + \sqrt{x^2 - a^2}| + C}$$Integration by Substitution
Method
If $\int f(x)\,dx$ is difficult, substitute $x = g(t)$:
$$\int f(x)\,dx = \int f(g(t)) \cdot g'(t)\,dt$$Common Substitutions
| Expression | Substitution |
|---|---|
| $\sqrt{a^2 - x^2}$ | $x = a\sin\theta$ |
| $\sqrt{a^2 + x^2}$ | $x = a\tan\theta$ |
| $\sqrt{x^2 - a^2}$ | $x = a\sec\theta$ |
| $a + b\cos x$ or $a + b\sin x$ | $\tan\frac{x}{2} = t$ |
Universal Substitution
For $\int R(\sin x, \cos x)\,dx$, use $\tan\frac{x}{2} = t$:
$$\sin x = \frac{2t}{1+t^2}, \quad \cos x = \frac{1-t^2}{1+t^2}, \quad dx = \frac{2\,dt}{1+t^2}$$Integration by Parts
$$\boxed{\int u\,dv = uv - \int v\,du}$$ILATE Rule
Choose $u$ in order of priority:
- Inverse trigonometric
- Logarithmic
- Algebraic
- Trigonometric
- Exponential
Special Cases
$$\int e^x[f(x) + f'(x)]\,dx = e^x f(x) + C$$Partial Fractions
For $\frac{P(x)}{Q(x)}$ where degree of $P <$ degree of $Q$:
Case 1: Linear Factors
$$\frac{1}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}$$Case 2: Repeated Linear Factors
$$\frac{1}{(x-a)^2(x-b)} = \frac{A}{x-a} + \frac{B}{(x-a)^2} + \frac{C}{x-b}$$Case 3: Quadratic Factors
$$\frac{1}{(x^2+a^2)(x-b)} = \frac{Ax+B}{x^2+a^2} + \frac{C}{x-b}$$Definite Integrals
Definition (Fundamental Theorem)
$$\boxed{\int_a^b f(x)\,dx = F(b) - F(a)}$$where $F'(x) = f(x)$
Properties
$\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$
$\int_a^b f(x)\,dx = \int_a^c f(x)\,dx + \int_c^b f(x)\,dx$
$\int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx$
$\int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx$
$\int_0^{2a} f(x)\,dx = \int_0^a f(x)\,dx + \int_0^a f(2a-x)\,dx$
Even and Odd Functions
For even function $f(-x) = f(x)$:
$$\int_{-a}^a f(x)\,dx = 2\int_0^a f(x)\,dx$$For odd function $f(-x) = -f(x)$:
$$\int_{-a}^a f(x)\,dx = 0$$Periodic Functions
If $f(x)$ has period $T$:
$$\int_0^{nT} f(x)\,dx = n\int_0^T f(x)\,dx$$Area Under Curves
Basic Formula
Area between curve $y = f(x)$, x-axis, and lines $x = a$, $x = b$:
$$A = \int_a^b |f(x)|\,dx$$Area Between Two Curves
$$A = \int_a^b |f(x) - g(x)|\,dx$$Standard Areas
| Curve | Area |
|---|---|
| Circle: $x^2 + y^2 = a^2$ | $\pi a^2$ |
| Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ | $\pi ab$ |
| Parabola: $y^2 = 4ax$ from 0 to $x$ | $\frac{4}{3}x\sqrt{ax}$ |
Practice Problems
Evaluate: $\int \frac{x^2 + 1}{x^4 + 1}\,dx$
Find: $\int_0^{\pi/2} \frac{\sin x}{\sin x + \cos x}\,dx$
Calculate: $\int e^x(\tan x + \sec^2 x)\,dx$
Find the area bounded by $y = x^2$ and $y = x$.