Free online courseMath Fundamentals Integrals

Conteúdo do Curso

• Video class: Primitive functions of the powers | 1/20 | UPV

  0h07m

• Exercise: Which of the following is the proper form of the antiderivative for the function x raised to a rational exponent, such as x^(n/m)?

• Video class: Linearity and primitive of polynomials | 2/20 | UPV

  0h05m

• Exercise: Given the polynomial function P(x) = 4x^3 - 5x + 1, what is the antiderivative, or the integral, of this function?

• Video class: Arctangent function and its derivative | 3/20 | UPV

  0h04m

• Exercise: What is the integral of the function 1 / (1 + x²) with respect to x?

• Video class: Pseudo-inmediate primitives | 4/20 | UPV

  0h06m

• Exercise: What is the integral of f(x) raised to 'a', given f(x) is a differentiable function?

• Video class: Integration I | 5/20 | UPV

  0h03m

• Exercise: Consider the integral \( \int (3x^2 + x) \cdot (x^3 + x + 1)^{\frac{1}{2}} \, dx \). What is the pseudo-immediate primitive of this integral?

• Video class: Integration II | 6/20 | UPV

  0h04m

• Exercise: Evaluate the integral \( \int \frac{3x}{x^2 + 2} \, dx \). Which method is most suitable for solving it?

• Video class: Integration III | 7/20 | UPV

  0h03m

• Exercise: Consider the integral of the function f(x) = x / (x^4 + 1). Which of the following substitutions would be most appropriate to simplify and solve this integral using the technique of substitution?

• Video class: Primitives of trigonometric functions | 8/20 | UPV

  0h05m

• Exercise: What is the antiderivative of the sine function?

• Video class: Completing the square | 9/20 | UPV

  0h07m

• Exercise: For the quadratic function f(x) = 2x^2 + 8x + 6, what is the vertex form of this parabola after completing the square?

• Video class: Arctangent primitives | 10/20 | UPV

  0h05m

• Exercise: What is the first step when integrating an arctangent type integral where the denominator is an irreducible second-degree polynomial?

• Video class: Primitives of rational functions I | 11/20 | UPV

  0h06m

• Exercise: Which of the following represents a correct primitive (antiderivative) of the function f(x) = 1/(x^2 - 1)?

• Video class: Primitives of rational functions II | 12/20 | UPV

  0h07m

• Exercise: Which integral technique is primarily used to decompose a rational function into simpler fractions for easier integration?

• Video class: Areas enclosed by a curve | 13/20 | UPV

  0h05m

• Exercise: Given the function f(x) is defined and positive over the interval [a, b], and F(x) is a primitive of f(x), what does the definite integral of f(x) from a to b represent?

• Video class: Area calculation (Single function) | 14/20 | UPV

  0h06m

• Exercise: Consider a function g(x) that is positive between x = 0 and x = 3 and negative between x = 3 and x = 5. What is the correct way to calculate the total area between the graph of g(x) and the x-axis over the interval [0, 5]?

• Video class: Area calculation (Two functions) | 15/20 | UPV

  0h07m

• Exercise: Consider two continuous functions, f(x) and g(x), where f(x) is above g(x) on the interval [a, b] and g(x) is above f(x) on the interval [b, c]. What is the correct way to calculate the area enclosed between these two functions from a to c?

• Video class: Area calculation (Complex I) | 16/20 | UPV

  0h05m

• Exercise: What is the total area of the region bounded by the lines x + y = 2, y = 2, and the parabola y = x² that includes the point (0, 1)?

• Video class: Area calculation (Complex II) | 17/20 | UPV

  0h04m

• Exercise: Consider the region bounded by the parabola y = x^2, the line y = 2, and the line x + y = 2. Calculate the area of this region.

• Video class: Change of variable | 18/20 | UPV

  0h07m

• Exercise: What is the primary purpose of using the change of variable technique in integral calculus?

• Video class: Integration by parts | 19/20 | UPV

  0h06m

• Exercise: When using integration by parts for the integral \( \int x \cdot e^x \, dx \), which functions should be chosen as 'u' and 'dv'?

• Video class: Practical applications | 20/20 | UPV

  0h08m

• Exercise: What is a key technique used to simplify the integral of compositions like the sine of the natural logarithm of x?