ملاحظات دراسة حساب التفاضل والتكامل

Calculus Study Notes

• Integration: The inverse process of differentiation, denoted by ∫. The symbol dx represents integration with respect to x, similarly dy represents integration with respect to y.

• Indefinite Integrals: Rules for calculating integrals without specific limits. The general form is ∫ f(x) dx = F(x) + C, where F(x) is the antiderivative of f(x) and C is the constant of integration.

• Rules: There are general rules for integration, but there is no singular rule for every function (fractions, roots, products of functions). A major type of integral rule is for integral of a constant.

• Example Integrals (with Constants):

• ∫ a dx = ax + c

• ∫ 6 dx = 6x + c

• ∫ √5 dx = √5x + c

• ∫ xⁿ dx = xn+1 / (n+1) + c (n ≠ -1)

• Power Rule for Integration: ∫x²dx = x³/3+C

• Example Integrals (with Variable):

• ∫x²dx = x³/3 + c

• ∫x⁵dx = x⁶/6+c

• ∫5x³dx = (5x⁴)/4 + c

• ∫4x²dx = (4x³/3) + c

• Integration of Roots: ∫√x dx = (2/3)x^(3/2) + C.

• Integration of Fractions: Example, ∫x⁻²dx = -x⁻¹ + C.

• Integration of Polynomial Functions: Examples for integrating polynomials

• ∫ (5x² + 3x - 2) dx = (5x³/3) + (3x²/2) - 2x + c

• Integration of Powers: Example,

• ∫ (3x² + 5)³(3x + 4) dx = (3x² + 5)⁴/8 + C

• Integration by substitution: Example, ∫2x√(x² + 8x + 5)(6x + 8) dx = (3x² + 8x + 5)⁷/7+c

• Integral of constants: A constant value multiplied by the variable will integrate to be the constant multiplied by the variable raised to an additional 1 power divided by the added power, plus the constant of integration, as in ∫(5x³)dx = 5x⁴/4 + C

• Definite Integrals: Integrals with specific limits of integration. The result represents the signed area under the curve between the limits a and b (∫a to b f(x)dx). Important: A constant of integration is not introduced in results of definite integration.