91Ó°ÊÓ

Trigonometric substitution is a technique used in calculus to simplify integrals. It involves replacing a variable with a trigonometric function to use trigonometric identities for simplification. This method is particularly useful for integrals involving square roots of quadratic expressions, such as the form \(\sqrt{u^{2}-a^{2}}\).

In the example provided, the integral to simplify is \(\int \frac{\mathrm{d} u}{u \sqrt{u^{2}-a^{2}}}\). Here, we recognize that by using the trigonometric identity \(\sec^2 \theta - 1 = \tan^2 \theta\), we can choose substitution to simplify the integral.

• Set \(u = a \sec \theta\)
• So, \(\mathrm{d} u = a \sec \theta \tan \theta \mathrm{d} \theta\)

This substitution helps us convert our integral into a simpler trigonometric integral, paving the way towards the solution.

An integral can be categorized as either definite or indefinite:

• Indefinite integrals represent a family of functions and include a constant of integration \(C\). It looks like \(\int f(x) \, \mathrm{d} x = F(x) + C\), where \(F(x)\) is the antiderivative of \(f(x)\).
• Definite integrals, on the other hand, have upper and lower limits. They give us a real number representing the area under the curve between those limits. It’s denoted as \(\int_{a}^{b} f(x) \, \mathrm{d} x = F(b) - F(a)\).

The integral given in the exercise, \(\int \frac{\mathrm{d} u}{u \sqrt{u^{2}-a^{2}}}\), is an indefinite integral. Solving it involves finding the antiderivative of the function inside the integral and including a constant of integration \(C\). Through substitution and simplification, we ultimately obtain the indefinite form \( \frac{1}{a} \sec ^{-1}\left(\frac{u}{a}\right) + C \).

Several calculus techniques are employed while solving integrals, and they considerably streamline the process:

• Substitution: This technique, also known as u-substitution, simplifies an integral by changing the variable of integration.
• Parts: Integration by parts is derived from the product rule of differentiation and helps in integrating products of functions.
• Partial Fractions: This technique is used to break down complex rational functions into simpler fractions that can be easily integrated.

In the provided solution, we mainly use trigonometric substitution, which simplifies integrals involving \(\sqrt{u^{2}-a^{2}}\). The steps include:

• Substituting \(u = a \sec \theta\) transforms the integral to a simpler form.
• Simplifying the trigonometric expressions to make integration feasible.
• Reverting back to the original variable using back substitution.

This step-by-step approach is critical for mastering various integral calculus problems.