01
Trigonometric Integrals
All Trig Standard Forms
F1
\(\displaystyle\int \sin x\,dx = -\cos x+C\)
F2
\(\displaystyle\int \cos x\,dx = \sin x+C\)
F3
\(\displaystyle\int \tan x\,dx = \ln|\sec x|+C\)
F4
\(\displaystyle\int \cot x\,dx = \ln|\sin x|+C\)
F5
\(\displaystyle\int \sec x\,dx = \ln|\sec x+\tan x|+C\)
F6
\(\displaystyle\int \csc x\,dx = \ln|\csc x-\cot x|+C\)
F7
\(\displaystyle\int \sec^2 x\,dx = \tan x+C\)
F8
\(\displaystyle\int \csc^2 x\,dx = -\cot x+C\)
F9
\(\displaystyle\int \sec x\tan x\,dx = \sec x+C\)
F10
\(\displaystyle\int \csc x\cot x\,dx = -\csc x+C\)
02
Algebraic Standard Forms
Forms Involving x² ± a²
A1 — 1/(x²−a²)
\(\displaystyle\int \frac{dx}{x^2-a^2} = \frac{1}{2a}\ln\!\left|\frac{x-a}{x+a}\right|+C\)
A2 — 1/(a²−x²)
\(\displaystyle\int \frac{dx}{a^2-x^2} = \frac{1}{2a}\ln\!\left|\frac{a+x}{a-x}\right|+C\)
A3 — 1/(x²+a²)
\(\displaystyle\int \frac{dx}{x^2+a^2} = \frac{1}{a}\tan^{-1}\!\frac{x}{a}+C\)
Forms Involving √(x² ± a²) and √(a² − x²)
B1 — 1/√(x²−a²)
\(\displaystyle\int \frac{dx}{\sqrt{x^2-a^2}} = \ln\!\left|x+\sqrt{x^2-a^2}\right|+C\)
B2 — 1/√(x²+a²)
\(\displaystyle\int \frac{dx}{\sqrt{x^2+a^2}} = \ln\!\left|x+\sqrt{x^2+a^2}\right|+C\)
B3 — 1/√(a²−x²)
\(\displaystyle\int \frac{dx}{\sqrt{a^2-x^2}} = \sin^{-1}\!\frac{x}{a}+C\)
Square Root Integrals
C1 — √(x²−a²)
\(\displaystyle\int \!\sqrt{x^2-a^2}\,dx = \frac{x\sqrt{x^2-a^2}}{2}-\frac{a^2}{2}\ln|x+\sqrt{x^2-a^2}|+C\)
C2 — √(x²+a²)
\(\displaystyle\int \!\sqrt{x^2+a^2}\,dx = \frac{x\sqrt{x^2+a^2}}{2}+\frac{a^2}{2}\ln|x+\sqrt{x^2+a^2}|+C\)
C3 — √(a²−x²)
\(\displaystyle\int \!\sqrt{a^2-x^2}\,dx = \frac{x\sqrt{a^2-x^2}}{2}+\frac{a^2}{2}\sin^{-1}\!\frac{x}{a}+C\)
Memory Tip: For \(\int\frac{dx}{x^2-a^2}\), recall the partial fraction: \(\frac{1}{x^2-a^2} = \frac{1}{2a}\!\left(\frac{1}{x-a}-\frac{1}{x+a}\right)\). Integrating gives the log form directly.
03
Integrals of Special Forms
Reducible Quadratic Forms — ax² + bx + c
Complete the square in the denominator to reduce to a standard form:
COMPLETE THE SQUARE TRICK
\[ ax^2+bx+c = a\!\left[\left(x+\frac{b}{2a}\right)^2 + \left(c - \frac{b^2}{4a}\right)\!\cdot\frac{1}{a}\right] \]
Then use \(t = x+\tfrac{b}{2a}\) and apply forms A1/A2/A3 or B1/B2/B3 as appropriate.
Form: (px + q) / (ax² + bx + c)
Split the numerator as: \(px+q = \lambda\cdot\frac{d}{dx}(ax^2+bx+c) + \mu\)
DECOMPOSITION STRATEGY
\[ \int \frac{px+q}{ax^2+bx+c}\,dx = \lambda\ln|ax^2+bx+c|+\mu\int\frac{dx}{ax^2+bx+c} \]
Find \(\lambda\) and \(\mu\) by comparing coefficients.
Form: (px + q) / √(ax² + bx + c)
Same split strategy as above, but with a square root in the denominator:
\[ \int \frac{px+q}{\sqrt{ax^2+bx+c}}\,dx = \lambda\sqrt{ax^2+bx+c}+\mu\int\frac{dx}{\sqrt{ax^2+bx+c}} \]
04
Quick Reference Table
| # | Integrand \(f(x)\) | Integral \(\int f(x)\,dx\) |
|---|---|---|
| 1 | \(x^n,\; n\ne-1\) | \(\dfrac{x^{n+1}}{n+1}+C\) |
| 2 | \(\dfrac{1}{x}\) | \(\ln|x|+C\) |
| 3 | \(e^x\) | \(e^x+C\) |
| 4 | \(a^x\) | \(\dfrac{a^x}{\ln a}+C\) |
| 5 | \(\sin x\) | \(-\cos x+C\) |
| 6 | \(\cos x\) | \(\sin x+C\) |
| 7 | \(\tan x\) | \(\ln|\sec x|+C\) |
| 8 | \(\cot x\) | \(\ln|\sin x|+C\) |
| 9 | \(\sec^2 x\) | \(\tan x+C\) |
| 10 | \(\csc^2 x\) | \(-\cot x+C\) |
| 11 | \(\dfrac{1}{x^2+a^2}\) | \(\dfrac{1}{a}\tan^{-1}\!\dfrac{x}{a}+C\) |
| 12 | \(\dfrac{1}{x^2-a^2}\) | \(\dfrac{1}{2a}\ln\!\left|\dfrac{x-a}{x+a}\right|+C\) |
| 13 | \(\dfrac{1}{\sqrt{a^2-x^2}}\) | \(\sin^{-1}\!\dfrac{x}{a}+C\) |
| 14 | \(\dfrac{1}{\sqrt{x^2+a^2}}\) | \(\ln\!\left|x+\sqrt{x^2+a^2}\right|+C\) |
| 15 | \(\dfrac{1}{\sqrt{x^2-a^2}}\) | \(\ln\!\left|x+\sqrt{x^2-a^2}\right|+C\) |
| 16 | \(\sqrt{a^2-x^2}\) | \(\dfrac{x\sqrt{a^2-x^2}}{2}+\dfrac{a^2}{2}\sin^{-1}\!\dfrac{x}{a}+C\) |
| 17 | \(\sqrt{x^2+a^2}\) | \(\dfrac{x\sqrt{x^2+a^2}}{2}+\dfrac{a^2}{2}\ln\!\left|x+\sqrt{x^2+a^2}\right|+C\) |
| 18 | \(\sqrt{x^2-a^2}\) | \(\dfrac{x\sqrt{x^2-a^2}}{2}-\dfrac{a^2}{2}\ln\!\left|x+\sqrt{x^2-a^2}\right|+C\) |
| 19 | \(e^x[f(x)+f'(x)]\) | \(e^x\cdot f(x)+C\) |
Exam Tip: Formulas 11–18 are frequently tested in board exams. Memorise them as a group — notice how they pair up: x²+a² pairs with arctan, and the square root forms pair with arcsin or ln.