Calculus 2_1stSem_2024-2025_BSCPE2A

Integral Calculus: A Brief Summary

  1. What Is an Integral?

    • Integrals are like cosmic calculators. They assign numbers to functions, describing displacement, motion, area, and volume.
    • The process of finding the anti-derivative (or integral) of a function is called integration. It’s the inverse of differentiation.
    • Integrals represent a family of curves—a universe of possibilities.

  2. Definite vs. Indefinite Integrals:

    • Definite integral: Calculates the area under a curve between specific limits (from (a) to (b)).
    • Indefinite integral: Finds the general antiderivative of a function (represented by (\int f(x) , dx)).

  3. Fundamental Theorems:

    • First Fundamental Theorem: If (A(x) = \int_a^x f(t) , dt), then (A’(x) = f(x)).
    • Second Fundamental Theorem: If (F(x)) is an antiderivative of (f(x)), then (\int_a^b f(x) , dx = F(b) - F(a)).

  4. Methods and Formulas:

    • Power Rule: (\int x^n , dx = \frac{1}{n+1} x^{n+1} + C)
    • Trigonometric Integrals: (\int \sin(x) , dx = -\cos(x) + C), (\int \cos(x) , dx = \sin(x) + C)
    • Exponential and Logarithmic Integrals: (\int e^x , dx = e^x + C), (\int \frac{1}{x} , dx = \ln|x| + C)

  5. Applications:

    • Area: Compute the area under curves.
    • Physics: Calculate work, displacement, and more.
    • Engineering: Model real-world phenomena.

  6. Remember the “+C”!

    • The constant of integration ((+C)) accounts for all possible antiderivatives.

And there you have it—a glimpse into the magical world of integrals!