91Ó°ÊÓ

Integration by parts is a technique used to integrate products of functions, drawing on the fundamental theorem of calculus. It helps in situations where a direct integral is not immediately obvious or feasible.
The formula for integration by parts is given as:
\[ \int u \, dv = uv - \int v \, du. \]
The strategy here is to identify parts of the integral that can be set as \(u\) and \(dv\). By choosing \(u = \sin^{-1}(x)\) and \(dv = dx\), the exercise leverages the ability to differentiate \(\sin^{-1}(x)\) while integrating \(dx\), simplifying the problem.

• Differentiate \(u\): \(du = \frac{1}{\sqrt{1-x^2}} \, dx\).
• Integrate \(dv\): \(v = x\).

These steps allow us to rewrite the original integral into a more manageable form. After applying the integration by parts formula, the focus shifts to solving a new integral, paving the way to a solution.

Inverse trigonometric functions, like \(\sin^{-1}(x)\), are essential when reversing trigonometric processes. They serve as the bridges in calculus for functions that relate angles and ratios. Notably, \(\sin^{-1}(x)\) signifies the angle whose sine is \(x\).
Understanding these functions' derivatives is crucial for integration by parts:

• The derivative of \(\sin^{-1}(x)\) is \(\frac{1}{\sqrt{1-x^2}}\).
• This derivative emerges from the trigonometric identity \(\sin^2(\theta) + \cos^2(\theta) = 1\).

When integrated, these derivatives allow for finding the antiderivative of inverse trigonometric functions, making them indispensable tools in the context of calculus and integration techniques.

Substitution is a technique that simplifies integration by making a substitution that transforms the integral into an easier form.
To tackle the remaining integral \(\int x \frac{1}{\sqrt{1-x^2}} \, dx\), the substitution \(w = 1-x^2\) effectively reduces complexity.

• Compute \(dw = -2x \, dx\), leading to \(x \, dx = -\frac{1}{2} \, dw\).
• This transforms the integral into \(-\frac{1}{2} \int w^{-1/2} \, dw\).

This new form makes it straightforward to integrate using power rules for integrals.
Ultimately, the integration yields \(-\sqrt{w} = -\sqrt{1-x^2}\), allowing subsequent back-substitution to complete the integration process.
Such strategic substitutions are powerful in recalibrating challenging integrals into more solvable forms, enhancing problem-solving efficiency.