How do you integrate a polynomial such as 4x^3 - 5x + 1?

Apply the power rule term by term: ∫(4x^3 - 5x + 1)dx = x^4 - (5/2)x^2 + x + C.

Why is completing the square useful when integrating functions with quadratic denominators?

It rewrites a quadratic into a form involving u² + a², which can lead directly to an arctangent antiderivative.

How is the total area between a curve and the x-axis calculated when the function changes sign?

Split the integral at each x-axis crossing and add the absolute values of the signed integrals.

Free Course Image Math Fundamentals Integrals

Free online course Math Fundamentals Integrals

Duration of the online course: 1 hours and 57 minutes

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Build confidence with integrals in this free online course: master antiderivatives, substitution, areas under curves, and integration by parts with practice.

In this free course, learn about

Apply power rule for antiderivatives, including rational exponents x^(n/m).
Use linearity to integrate polynomials and sums of basic functions.
Recognize and integrate 1/(1+x^2) as arctan(x)+C; link arctan and its derivative.
Identify pseudo-immediate primitives using pattern f'(x)*(f(x))^a and substitution.
Choose effective substitutions for rational forms like x/(x^4+1) and 3x/(x^2+2).
Integrate basic trigonometric functions (e.g., ∫sin x dx = -cos x + C).
Complete the square to rewrite quadratics and prepare arctan-type integrals.
Integrate arctan-type forms with irreducible quadratics after completing the square.
Integrate rational functions using partial fraction decomposition and standard primitives.
Interpret definite integrals as (signed) area; compute total area using absolute values.
Compute area between curves with piecewise top-minus-bottom (or right-minus-left) integrals.
Solve complex bounded-area problems by finding intersections and splitting integrals.
Apply change of variable to simplify integrals of compositions like sin(ln x).
Use integration by parts (choose u and dv) for products like ∫x e^x dx.

About the free online course

Strengthen your calculus skills by learning how integrals work from the inside out, with an approach that connects clear intuition to reliable techniques. This free online course focuses on the fundamentals of integration: how to recognize patterns, choose an efficient method, and verify results with confidence. If derivatives feel familiar but antiderivatives still seem like a puzzle, this training helps you make the transition from rules you memorize to procedures you actually understand and can apply.

You will practice finding primitive functions for powers, polynomials, trigonometric expressions, and classic forms that lead to logarithms or arctangent results. Along the way, you will develop a sense for linearity and how small algebraic changes simplify a problem before you start integrating. You will also learn to spot pseudo-immediate situations where the structure of a function and its derivative appear together, making substitution feel natural rather than forced.

As the course progresses, integration becomes more than symbol manipulation. You will work with rational functions and the ideas behind rewriting them into simpler pieces so that each part becomes integrable. You will also revisit completing the square to prepare quadratics for standard integral forms, which is especially useful when expressions do not factor nicely. These habits build the problem-solving reflexes that students need in geometry-related applications, where curves, areas, and changing shapes are central.

A major payoff of integration is computing area. You will learn how definite integrals represent accumulated quantities, how to handle regions where a function changes sign, and how to find the area between two curves when one overtakes the other. You will also tackle more complex bounded regions involving lines and parabolas, building the geometric interpretation that makes results feel meaningful rather than arbitrary.

To round out your toolkit, you will explore change of variable and integration by parts, including how to choose good components so the method simplifies instead of complicating the work. By the end, you should be able to approach unfamiliar integrals with a plan, select an appropriate strategy, and connect the answer back to the shape of the graph or the quantity being measured.

Course content

Video class: Primitive functions of the powers | 1/20 | UPV 07m
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 05m
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 04m
Exercise: What is the integral of the function 1 / (1 + x²) with respect to x?
Video class: Pseudo-inmediate primitives | 4/20 | UPV 06m
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 03m
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 04m
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 03m
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 05m
Exercise: What is the antiderivative of the sine function?
Video class: Completing the square | 9/20 | UPV 07m
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 05m
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 06m
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 07m
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 05m
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 06m
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 07m
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 05m
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 04m
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 07m
Exercise: What is the primary purpose of using the change of variable technique in integral calculus?
Video class: Integration by parts | 19/20 | UPV 06m
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 08m
Exercise: What is a key technique used to simplify the integral of compositions like the sine of the natural logarithm of x?