Integration by Introduction of a New Variable 1 (Substitution Method)

INTEGRATION BY INTRODUCTION OF A NEW VARIABLE

Integration by introduction of a new variable, also known as the substitution method (or u-substitution), is a key technique in integral calculus. This approach allows for the simplification of complex integrals that would otherwise be difficult to solve. The technique is based on replacing the original variable with a new one, which leads to a simpler form of the integral.

HOW DOES IT WORK?

The method requires us to choose an appropriate substitution for the variable, usually with the aim of reducing the complexity of the integral.

When integrating by introduction of a new variable (substitution method), the following steps are typically used:

1. CHOOSE A SUBSTITUTION: Find a suitable substitution, often in the form u = g(x) (the article uses f(x), but g(x) is also common to avoid confusion with the main function being integrated), that simplifies the given integral. The choice of 'u' is often a part of the integrand whose derivative is also present (or can be easily formed).
2. CALCULATE du: Calculate du as the derivative of g(x) with respect to x, and multiply by dx. So, if u = g(x), then du = g′(x)dx.
3. SUBSTITUTE INTO THE INTEGRAL: Replace all expressions in the integral involving x and dx with the corresponding expressions in terms of u and du. The entire integral should now be in terms of 'u'.
4. INTEGRATE WITH RESPECT TO u: Perform the integration with respect to the new variable 'u'.
5. RETURN TO THE ORIGINAL VARIABLE: After integration, substitute 'u' back in terms of 'x' using the original substitution u = g(x).

EXAMPLES OF USE

The substitution method is widely applicable in various mathematical contexts, including the integration of functions containing polynomials, trigonometric functions, exponential functions, and many others.

CONCLUSION

Integration by introduction of a new variable (substitution method) is a powerful tool in integral calculus. It allows us to tackle a wide range of integrals that would be difficult to solve without this method. The key to successfully using this method lies in the skill of choosing the right substitution that simplifies the problem to the extent that integration becomes feasible.