Shortcut Methods
Numerical of Indefinite Integral
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Power Rule: $$\int x^n dx = \frac{x^{n+1}}{n+1} + C$$ for any real number (n \neq -1).
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Sum Rule: $$\int (f(x) + g(x)) dx = \int f(x) dx+ \int g(x) dx $$
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Difference Rule: $$\int (f(x) - g(x)) dx = \int f(x) dx- \int g(x) dx$$
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Constant Multiple Rule: $$\int c f(x) dx = c \int f(x) dx$$
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Substitution Rule: If (u = g(x)) is a differentiable function, then $$\int f(g(x)) g’(x) dx = \int f(u) du $$
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Logarithmic rule: $$\int{\frac{1}{x}dx} = \ln{|x|}+C, (x \neq 0)$$
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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
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Area under a curve: $$Area = \int_{a}^{b} f(x) dx$$
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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 cross-section 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).
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Average value of a function: $$Average value = \frac{1}{b-a} \int_{a}^{b} f(x) dx$$
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Probability: $$Probability = \int_{a}^{b} f(x) dx,$$
where (f(x)) is the probability density function of the random variable (X).
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