Shortcut Methods
Numerical of Indefinite Integral

Power Rule: $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$ for any real number (n \neq 1).

Sum Rule: $$\int (f(x) + g(x)) dx = \int f(x) dx+ \int g(x) dx $$

Difference Rule: $$\int (f(x)  g(x)) dx = \int f(x) dx \int g(x) dx$$

Constant Multiple Rule: $$\int c f(x) dx = c \int f(x) dx$$

Substitution Rule: If (u = g(x)) is a differentiable function, then $$\int f(g(x)) g’(x) dx = \int f(u) du $$

Logarithmic rule: $$\int{\frac{1}{x}dx} = \ln{x}+C, (x \neq 0)$$

Exponential Rule: $$\int e^x dx = e^x + C$$
Numerical of Definite Integral
$$\int_{a}^{b} f(x) dx = F(b)  F(a),$$
where (F(x)) is an antiderivative of (f(x)) (i.e. (\frac{d}{dx}F(x) = f(x))).
Applications of Integral Calculus

Area under a curve: $$Area = \int_{a}^{b} f(x) dx$$

Volume of a solid generated by revolving a region around an axis: $$Volume = \int_{a}^{b} A(x) dx,$$
where (A(x)) is the area of the crosssection of the solid at (x).
 Work done by a force: $$Work = \int_{a}^{b} F(x) dx,$$
where (F(x)) is the force applied at the point (x).

Average value of a function: $$Average value = \frac{1}{ba} \int_{a}^{b} f(x) dx$$

Probability: $$Probability = \int_{a}^{b} f(x) dx,$$
where (f(x)) is the probability density function of the random variable (X).