Calculus
Class 12

Basic Integration - Power Rule

xndx=xn+1n+1+C (for n1)\int x^n dx = \frac{x^{n+1}}{n+1} + C \text{ (for } n \neq -1)
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∫xⁿdx = x^(n+1)/(n+1) + C

Adjust Variables

Exponent n
n =
04

Reverse of differentiation. The constant C accounts for any constant term lost during differentiation.

Real-World Applications

Physics — Distance from velocity: s = ∫v dt.

Economics — Total cost from marginal cost: TC = ∫MC dq.

Engineering — Area under stress-strain curve = energy absorbed.

Biology — Total population from growth rate.

Finance — Present value from cash flow rate.

Chemistry — Total amount from reaction rate.

Statistics — CDF from PDF: F(x) = ∫f(x)dx.

Electrical — Charge from current: Q = ∫I dt.

Mechanics — Work from force: W = ∫F dx.

Hydrology — Total flow from flow rate.

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