91Ó°ÊÓ

Trigonometric substitution is a powerful technique in integral calculus used to simplify integrals that involve square roots or expressions involving trigonometric identities. The essence is to replace the variable in the integral with a trigonometric function. This replacement leverages the Pythagorean identities, such as \( \sin^2(x) + \cos^2(x) = 1 \), to simplify the expression.
In the context of this exercise, the integral \( \int \frac{d\theta}{\sqrt{1 - (\theta-1)^2}} \) is tackled by using trigonometric substitution. By recognizing \( x = \theta - 1 \) and substituting \( x = \sin(u) \), we transform the integral into an expression involving \( \int du \), which is much simpler to evaluate.
This approach brings into play the arcsin function naturally, as \( 1 - \sin^2(u) \) becomes \( \cos^2(u) \) allowing for simplification. Trigonometric substitution is especially useful when integrating expressions of the form \( \sqrt{a^2 - x^2} \), \( \sqrt{a^2 + x^2} \), and \( \sqrt{x^2 - a^2} \).

• Identify part of the expression that resembles a trigonometric identity.
• Choose appropriate substitution based on the form.
• Simplify using known trigonometric identities.

Inverse trigonometric functions play a crucial role in calculus, particularly in solving integrals involving trigonometric substitutions. They help in expressing angles or values that originally came from trigonometric functions.
In the given problem, after simplifying the integral using trigonometric substitution, we end up with an integral \( \int du \), which directly integrates to \( u + C \). Since \( u \) was introduced as part of \( x = \sin(u) \), solving for \( u \) leads us to the function \( u = \arcsin(x) \), where \( x = \theta - 1 \). Thus, the final answer for the integral is written in terms of the inverse sine function:
\[ \arcsin(\theta - 1) + C \].The arcsin function is one of the six main inverse trigonometric functions, others being arccos, arctan, arccot, arcsec, and arccsc. They are crucial whenever reverting back from trigonometric substitutions.

• Understand how to move from a trigonometric function back to an angle.
• Recognize when a result will be in the form of an inverse trig function.
• Ensure proper back-substitution to solve for the original variable.

Completing the square is an algebraic method used to simplify quadratic expressions, allowing for easier integration, especially when coupled with trigonometric substitution.
In this problem, the expression \( 2\theta - \theta^2 \) under the square root is not in a convenient form. By rewriting as \( -((\theta - 1)^2 - 1) \), we transform it into something more familiar: \( 1 - (\theta-1)^2 \). This form is ideal for trig substitution as it resembles the Pythagorean identity \( 1 - x^2 = \sin^2(u) + \cos^2(u) \).
The process involves:

• Rewriting the quadratic expression in the form \((x-h)^2 + k\).
• Recognizing the new expression easily substitutes into an integral.
• Using the rearranged form to assist in subsequent substitutions or transformations.

Completing the square is a vital step for tackling integrals that involve square roots of quadratic expressions.