Integral Calculus Formula Sheet

One-page rapid revision of every integration formula, technique, definite-integral property, and area result from this chapter. Scan it the night before the exam.

Indefinite Integral: Definition

$$\boxed{\int f(x)\,dx = F(x) + C \quad \text{where } F'(x) = f(x)}$$

Standard Integrals

Power and Exponential

Function	Integral
$x^n,\ n \neq -1$	$\dfrac{x^{n+1}}{n+1} + C$
$\dfrac{1}{x}$	$\ln\lvert x\rvert + C$
$e^x$	$e^x + C$
$a^x,\ a>0,\ a\neq 1$	$\dfrac{a^x}{\ln a} + C$

$$\boxed{\int x^n\,dx = \frac{x^{n+1}}{n+1} + C\ (n\neq -1)}\qquad \boxed{\int \frac{1}{x}\,dx = \ln\lvert x\rvert + C}$$

Trigonometric (the Big 6)

Function	Integral
$\sin x$	$-\cos x + C$
$\cos x$	$\sin x + C$
$\sec^2 x$	$\tan x + C$
$\csc^2 x$	$-\cot x + C$
$\sec x \tan x$	$\sec x + C$
$\csc x \cot x$	$-\csc x + C$
$\tan x$	$\ln\lvert\sec x\rvert + C$
$\cot x$	$\ln\lvert\sin x\rvert + C$

Inverse-Trigonometric Forms

$$\int \frac{1}{\sqrt{1-x^2}}\,dx = \sin^{-1}x + C \qquad \int \frac{-1}{\sqrt{1-x^2}}\,dx = \cos^{-1}x + C$$$$\int \frac{1}{1+x^2}\,dx = \tan^{-1}x + C \qquad \int \frac{-1}{1+x^2}\,dx = \cot^{-1}x + C$$$$\int \frac{1}{\lvert x\rvert\sqrt{x^2-1}}\,dx = \sec^{-1}x + C$$

Special Integrals (Standard Forms)

$$\boxed{\int \frac{dx}{x^2 + a^2} = \frac{1}{a}\tan^{-1}\frac{x}{a} + C}$$$$\boxed{\int \frac{dx}{x^2 - a^2} = \frac{1}{2a}\ln\left\lvert\frac{x-a}{x+a}\right\rvert + C}$$$$\boxed{\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\frac{x}{a} + C}$$$$\boxed{\int \frac{dx}{\sqrt{x^2 + a^2}} = \ln\left\lvert x + \sqrt{x^2 + a^2}\right\rvert + C}$$$$\boxed{\int \frac{dx}{\sqrt{x^2 - a^2}} = \ln\left\lvert x + \sqrt{x^2 - a^2}\right\rvert + C}$$

Properties of Indefinite Integrals

Property	Formula
Constant multiple	$\int k\,f(x)\,dx = k\int f(x)\,dx$
Sum / difference	$\int [f(x)\pm g(x)]\,dx = \int f(x)\,dx \pm \int g(x)\,dx$

Technique 1: Substitution (Reverse Chain Rule)

$$\boxed{\int f(g(x))\,g'(x)\,dx = \int f(u)\,du,\quad u = g(x)}$$

Key pattern — derivative of denominator on top:

$$\boxed{\int \frac{f'(x)}{f(x)}\,dx = \ln\lvert f(x)\rvert + C}$$

Trigonometric Substitutions

Expression	Substitution
$\sqrt{a^2 - x^2}$	$x = a\sin\theta$
$\sqrt{a^2 + x^2}$	$x = a\tan\theta$
$\sqrt{x^2 - a^2}$	$x = a\sec\theta$
$a + b\sin x$ or $a + b\cos x$	$\tan\frac{x}{2} = t$

Weierstrass (Universal) Substitution

For $\int R(\sin x, \cos x)\,dx$, let $t = \tan\frac{x}{2}$:

$$\sin x = \frac{2t}{1+t^2}, \qquad \cos x = \frac{1-t^2}{1+t^2}, \qquad dx = \frac{2\,dt}{1+t^2}$$

Technique 2: Integration by Parts

$$\boxed{\int u\,dv = uv - \int v\,du}$$

Special result:

$$\boxed{\int e^x\big[f(x) + f'(x)\big]\,dx = e^x f(x) + C}$$

Reduction formula (by parts, applied repeatedly):

$$I_n = \int x^n e^x\,dx = x^n e^x - n\,I_{n-1}$$

Circular trick: $\int e^{ax}\sin bx\,dx$ and $\int e^{ax}\cos bx\,dx$ — apply by parts twice; the integral reappears, then solve algebraically. Example result:

$$\int e^x \sin x\,dx = \frac{e^x(\sin x - \cos x)}{2} + C$$

Technique 3: Partial Fractions

For a proper rational function ($\deg P < \deg Q$); if not proper, do long division first.

Denominator type	Decomposition
Distinct linear $(x-a)(x-b)$	$\dfrac{A}{x-a} + \dfrac{B}{x-b}$
Repeated linear $(x-a)^2(x-b)$	$\dfrac{A}{x-a} + \dfrac{B}{(x-a)^2} + \dfrac{C}{x-b}$
Irreducible quadratic $(x^2+a^2)(x-b)$	$\dfrac{Ax+B}{x^2+a^2} + \dfrac{C}{x-b}$

Trigonometric Integration

Power-Reduction / Pythagorean Identities

$$\boxed{\sin^2 x = \frac{1 - \cos 2x}{2}} \qquad \boxed{\cos^2 x = \frac{1 + \cos 2x}{2}}$$$$\boxed{\tan^2 x = \sec^2 x - 1} \qquad \boxed{\cot^2 x = \csc^2 x - 1}$$

Product-to-Sum

$$\sin A \cos B = \tfrac{1}{2}[\sin(A+B) + \sin(A-B)]$$$$\cos A \sin B = \tfrac{1}{2}[\sin(A+B) - \sin(A-B)]$$$$\cos A \cos B = \tfrac{1}{2}[\cos(A+B) + \cos(A-B)]$$$$\sin A \sin B = \tfrac{1}{2}[\cos(A-B) - \cos(A+B)]$$

Triple-Angle

$$\sin^3 x = \frac{3\sin x - \sin 3x}{4} \qquad \cos^3 x = \frac{3\cos x + \cos 3x}{4}$$

Must-Memorize Trig Integrals

$$\boxed{\int \sec x\,dx = \ln\lvert\sec x + \tan x\rvert + C}$$$$\boxed{\int \csc x\,dx = \ln\lvert\csc x - \cot x\rvert + C}$$

Reduction Formula for $\sin^n x$

$$I_n = \int \sin^n x\,dx = -\frac{\sin^{n-1}x\,\cos x}{n} + \frac{n-1}{n}\,I_{n-2}$$

Strategy Table

Integral	First step	Key identity
$\int \sin^m x\cos^n x\,dx$ ($m$ or $n$ odd)	Save one factor, substitute	$\sin^2 = 1-\cos^2$ / $\cos^2 = 1-\sin^2$
$\int \sin^m x\cos^n x\,dx$ (both even)	Power reduction	$\sin^2 x = \frac{1-\cos 2x}{2}$
$\int \tan^m x\sec^n x\,dx$ ($n$ even)	Peel $\sec^2 x$	$\sec^2 x = 1 + \tan^2 x$
$\int \tan^m x\sec^n x\,dx$ ($m$ odd)	Peel $\sec x\tan x$	$\tan^2 x = \sec^2 x - 1$
$\int \sin Ax\cos Bx\,dx$	Product-to-sum	Convert to sum of sines
$\int \frac{1}{a+b\sin x}\,dx$	Weierstrass / rationalize	$t = \tan(x/2)$

Definite Integrals

Fundamental Theorem of Calculus

$$\boxed{\int_a^b f(x)\,dx = F(b) - F(a) = \big[F(x)\big]_a^b}$$$$\boxed{\frac{d}{dx}\left[\int_a^x f(t)\,dt\right] = f(x)}$$

Properties

#	Property
1	$\int_a^b f(x)\,dx = -\int_b^a f(x)\,dx$
2	$\int_a^a f(x)\,dx = 0$
3	$\int_a^b f(x)\,dx + \int_b^c f(x)\,dx = \int_a^c f(x)\,dx$
4	$\int_a^b [k f(x) + g(x)]\,dx = k\int_a^b f(x)\,dx + \int_a^b g(x)\,dx$

Symmetry & King Property

$$\boxed{\int_0^a f(x)\,dx = \int_0^a f(a-x)\,dx} \qquad \boxed{\int_a^b f(x)\,dx = \int_a^b f(a+b-x)\,dx}$$

Even / Odd Functions (symmetric limits)

$$\boxed{\int_{-a}^a f(x)\,dx = 2\int_0^a f(x)\,dx \ \ (f \text{ even})} \qquad \boxed{\int_{-a}^a f(x)\,dx = 0 \ \ (f \text{ odd})}$$

Periodic Functions (period $T$)

$$\boxed{\int_0^{nT} f(x)\,dx = n\int_0^T f(x)\,dx} \qquad \int_a^{a+T} f(x)\,dx = \int_0^T f(x)\,dx$$

Standard Definite Results

$$\int_0^{\pi/2} \sin^n x\,dx = \int_0^{\pi/2} \cos^n x\,dx$$

Wallis’s formula:

$$\int_0^{\pi/2} \sin^{2n} x\,dx = \int_0^{\pi/2} \cos^{2n} x\,dx = \frac{(2n-1)(2n-3)\cdots 3\cdot 1}{(2n)(2n-2)\cdots 4\cdot 2}\cdot\frac{\pi}{2}$$$$\int_0^{\pi/2} \sin^{2n+1} x\,dx = \int_0^{\pi/2} \cos^{2n+1} x\,dx = \frac{(2n)(2n-2)\cdots 4\cdot 2}{(2n+1)(2n-1)\cdots 3\cdot 1}$$

Recurring King-property result (any power $n$):

$$\int_0^{\pi/2} \frac{\sin^n x}{\sin^n x + \cos^n x}\,dx = \frac{\pi}{4}$$

Leibniz Rule (variable limits)

$$\boxed{\frac{d}{d\alpha}\left[\int_{g(\alpha)}^{h(\alpha)} f(x)\,dx\right] = f(h(\alpha))\,h'(\alpha) - f(g(\alpha))\,g'(\alpha)}$$

Estimation Inequalities

If $f(x) \le g(x)$ on $[a,b]$: $\displaystyle\int_a^b f\,dx \le \int_a^b g\,dx$.

If $m \le f(x) \le M$ on $[a,b]$: $\displaystyle m(b-a) \le \int_a^b f(x)\,dx \le M(b-a)$.

Applications: Area

Region	Formula
Under $y=f(x)$, x-axis, $x=a$ to $b$	$A = \int_a^b f(x)\,dx$ (if $f\ge 0$)
Actual (signed) area	$A = \int_a^b \lvert f(x)\rvert\,dx$
Between curves, $f \ge g$	$A = \int_a^b [f(x)-g(x)]\,dx$
Horizontal strips, $x = g(y)$	$A = \int_c^d g(y)\,dy$
Between $x=g_2(y) \ge g_1(y)$	$A = \int_c^d [g_2(y)-g_1(y)]\,dy$
Parametric curve	$A = \int_\alpha^\beta y(t)\,x'(t)\,dt$

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 $y^2=4ax$, $0$ to $x$	$\frac{4}{3}x\sqrt{ax}$
One hump of $\sin x$ ($0$ to $\pi$)	$2$
One hump of $\cos x$ ($-\frac{\pi}{2}$ to $\frac{\pi}{2}$)	$2$
$\lvert x\rvert + \lvert y\rvert = 1$ (diamond)	$2$

Method Selection Cheat Sheet

graph TD
    A[Integral to solve] --> B{Matches a standard form?}
    B -->|Yes| C[Apply formula directly]
    B -->|No| D{Rational function?}
    D -->|Numerator = derivative of denom| E[ln of denominator]
    D -->|Proper fraction| F[Partial fractions]
    D -->|Improper| G[Long division first]
    A --> H{Product of functions?}
    H -->|One is derivative of other| I[Substitution]
    H -->|Unrelated functions| J[By parts - LIATE]
    A --> K{Trigonometric?}
    K -->|Powers of sin/cos/tan/sec| L[SOCR / power reduction]
    K -->|Rational in sin/cos| M[Weierstrass t = tan x/2]
    A --> N{Definite with symmetry?}
    N -->|Yes| O[POSEK properties]

Cross-Links

• Indefinite Integrals
• Integration Techniques
• Trigonometric Integration
• Definite Integrals
• Applications of Integrals