The Master Formula Box
Power Rule: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ (for $n \neq -1$)Log Rule: $\int \frac{1}{x} \, dx = \ln|x| + C$Exponential: $\int e^x \, dx = e^x + C$
Dimensional Analysis of Integrals
The units of an integral $\int y \, dx$ are always $[Unit_y \times Unit_x]$. In Physics, if your $y$-axis is Velocity $[LT^{-1}]$ and your $x$-axis is Time $[T]$, the integral has units of $[L]$ (Displacement). If you are integrating Pressure $[ML^{-1}T^{-2}]$ over Volume $[L^3]$, the result has units of Energy $[ML^2T^{-2}]$ (Work). Always check this!
Variations: Properties of Definite Integrals
- King's Property: $\int_a^b f(x) \, dx = \int_a^b f(a+b-x) \, dx$. (Extremely powerful for JEE Trig integrals).
- Odd/Even Property: If $f(x)$ is odd, $\int_{-a}^a f(x) \, dx = 0$.
- Additivity: $\int_a^c = \int_a^b + \int_b^c$. (Used for Absolute Value/Piecewise functions).
Shortcuts & Mnemonics
- ILATE: To remember the order of $u$ in Integration by Parts.
- The "Area is Positive" Rule: If a curve is below the x-axis, the integral will be negative. When asked for "Area," take the absolute value!
- Mnemonic: "Integrate to accumulate."
Edge Cases
- $n = -1$: The power rule fails. The integral is $\ln|x|$.
- Improper Integrals: Where one of the limits is $\infty$. This is solved using limits: $\lim_{t \to \infty} \int_a^t f(x) \, dx$.
- Discontinuity: If $f(x)$ is not continuous on $[a, b]$, the integral might not exist or must be split at the point of discontinuity.
