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The substitution method is a fundamental technique in integration that simplifies complex integrals. It involves changing the variable of integration to transform a problem into a more manageable form. This often utilizes a substitution that aligns with the trigonometric identity or simplifies the integrand.

For instance, in the given exercise, we used the substitution \( x = \sin t \). This step converts the expression under the integral into terms of \( t \), making it simpler to integrate. This works well when the integrand contains expressions like \( \sqrt{1-x^2} \), which are recognizable as derivatives or related forms of trigonometric functions.

Here are some steps to effectively use the substitution method:

• Identify a substitution that might simplify the integral.
• Express the differential \( dx \) in terms of the new variable \( dt \).
• Substitute according to the chosen variable into the integral.
• Simplify and evaluate the integral with respect to the new variable.
• Finally, don't forget to substitute back to the original variable to get the result in terms of \( x \).

This method is powerful and often necessary for integrals involving trigonometric functions or their inverses.

Trigonometric integration is a technique employed to solve integrals involving trigonometric functions like sine, cosine, tangent, etc. In this exercise, the integral involves \( \sin^{-1} x \), which is an inverse trigonometric function. Recognizing how these functions interact with one another can simplify seemingly complex problems.

When substituting \( x = \sin t \), you effectively leverage the identity \( \sin^2 t + \cos^2 t = 1 \) to rewrite the integrand expressions. Notice how \( \sqrt{1-x^2} = \cos t \), following directly from the Pythagorean identity once \( x = \sin t \) is substituted. This substitution aligns the integral components to reveal a straightforward integration path.

Benefits of trigonometric integration include:

• Utilizing known identities to simplify complex functions into basic forms.
• Removing the square roots and complex terms using identities.
• Creating a direct path to integration by reducing complex forms into familiar trigonometric terms.

Understanding trigonometric identities thoroughly can simplify integration tasks and offer clearer solutions.

Inverse trigonometric functions, such as \( \sin^{-1} x \), \( \cos^{-1} x \), and \( \tan^{-1} x \), often appear in integration problems. They represent angles whose trigonometric function equals a given number. In our exercise, the function \( \sin^{-1} x \) is integrated, requiring understanding both its definition and derivatives.

In mathematical terms, for \( y = \sin^{-1} x \), the relationship is \( x = \sin y \), implying that \( y \) is the angle whose sine is \( x \). This understanding is crucial when differentiating or integrating these functions.

Here are some key points about inverse trigonometric functions to remember:

• They have specific derivatives which become useful during integration and differentiation tasks, e.g., \( \frac{d}{dx}[\sin^{-1} x] = \frac{1}{\sqrt{1-x^2}} \).
• They translate angles measured as ratios into values within specific ranges: for \( \sin^{-1} x \), the range is \([-\frac{\pi}{2}, \frac{\pi}{2}]\).
• Integrals involving these functions typically benefit from initial substitutions and expressions that match their derivative forms.

Mastering these aspects can further simplify integration challenges, providing clearer paths from complex integrands to simplified results.