Integrating Expressions Using Standard Results

Understanding Basic Forms of Integration

Become familiar with the concept of integration, a fundamental process in calculus, which can be seen as the inverse operation to differentiation.
Understand that standard results of integration are simply standard expressions that have known antiderivatives. These results are often used as a shortcut to achieve the antiderivative of a complex expression.
Remember that the basic forms of integrated expressions are notated as ∫f(x) dx, where f(x) is the function being integrated and dx is a small difference in x.

Learn standard results for integration such as ∫dx = x + C, ∫x^n dx = [x^(n+1)]/(n+1) + C, ∫sin(x) dx = -cos(x) + C, ∫cos(x) dx = sin(x) + C and many more.
Remember that in all these results, ‘n’ cannot be equal to -1; in case ‘n’ is -1, the standard result ∫x^-1 dx = ln x + C should be used.
Learn that ‘C’ is the constant of integration which is added whenever you integrate a function, it represents a possible family of solutions.

Understand the significance of standard results in simplifying a complex integration problem. They help to fast-track the computation by providing the antiderivatives of standard functions.
Practice using the standard results to integrate various functions like polynomials, trigonometric functions, exponential functions, and logarithmic functions.
Master the skill of identifying the best standard result to apply in a given integration problem to simplify the work involved.

Regularly work through various problems and examples varying from straightforward ones to complex ones, to enhance your skill in applying standard results in integration.
Utilise the solutions given to grasp the step-by-step application of each result and to verify the correctness your work.
Be aware of the common mistakes that may occur during the integration process and learn how to avoid them.

Learn to appreciate the practical applications of standard results in integration, mostly in fields such as physics, engineering and economics where they are used to solve real-world problems.
Understand how standard results in integration can be utilised in solving problems involving rates of change, area under a curve, and in deriving certain mathematical models.