Calculus Shortcut Formulas: Solve Definite Integrals in Under 30 Seconds

Why Standard Integration Takes Too Long

In Dhaka University 'Ka' unit and standard engineering MCQ exams, you have less than 45 seconds per math question. Writing 10 lines of integration by parts is suicide.

1. King's Property of Definite Integrals

$$\int_{a}^{b} f(x) dx = \int_{a}^{b} f(a + b - x) dx$$
When you apply this to integrals involving $\sin^n x / (\sin^n x + \cos^n x)$ over $[0, \pi/2]$, the answer is always $\frac{b - a}{2} = \frac{\pi}{4}$ without calculating a single antiderivative!

2. Wallis Formula for Powers of Sine and Cosine

For $\int_0^{\\pi/2} \sin^n x dx$, evaluate down the odd and even product sequences. It converts complex trigonometric integrals into elementary fraction multiplications.

3. Odd and Even Function Symmetry

Always inspect the limits. If the interval is $[-a, a]$ and the integrand is odd ($f(-x) = -f(x)$), the integral is identically 0!