91Ó°ÊÓ

The substitution method is a powerful technique in integral calculus. It helps simplify complex integrals by changing variables. The core idea is similar to the chain rule in differentiation. We choose a substitution that makes the integral easier to evaluate.

In our problem, we use the substitution \( u = \tan x \). This choice is strategic because it simplifies both the numerator and the denominator of the integral. We also find that \( du = \sec^2 x \, dx \), which indicates the differential change. By replacing \( \tan x \) with \( u \) and handling the differential correctly, we transform the given integral into a more manageable form:

• Replace \( \tan x \) with \( u \).
• \( \sec^2 x \, dx \) becomes \( du \).
• The expression \( \sqrt{1-\tan^2 x} \) changes to \( \sqrt{1-u^2} \).

The substitution method helped us change a complex trigonometric integrand into a simpler algebraic form.

Inverse trigonometric functions are used to find angles when the trigonometric ratios are known. They are often encountered in integration as they simplify integrals involving square roots.

After performing the substitution and simplifying the integrand in our problem, we end up with the integral \( \int \frac{1}{\sqrt{1-u^2}} \, du \). This is a well-known integral that corresponds to an inverse trigonometric function.

• \( \int \frac{1}{\sqrt{1-u^2}} \, du \) is recognizable as \( \arcsin u + C \).

This result shows that the indefinite integral evaluates to the inverse sine of \( u \). Understanding these forms allows one to quickly identify integrals as standard results of inverse trigonometric functions.

Standard integral forms are crucial in recognizing integrals that conform to these patterns. They enable quick and accurate integral evaluation without having to perform lengthy calculations.

In the given exercise, after substitution, we identify the integral \( \int \frac{1}{\sqrt{1-u^2}} \, du \). This is a classic example of a standard integral form. The form \( \int \frac{1}{\sqrt{1-x^2}} \, dx \) is the integral of \( \arcsin x \).

• Having these patterns memorized helps speed up the integration process.
• It also minimizes mistakes since these are derived results.

By recognizing the standard integral form, one can confidently replace the integral expression with its known solution, leading to accurate and efficient problem-solving.