91Ó°ÊÓ

The power rule is one of the fundamental tools for integrating functions. It is especially useful for finding the indefinite integrals of polynomials of the form \( x^n \), where \( n \) is a real number. The rule states:

• \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)

The constant "\( C \)" represents the integration constant. Be cautious, as this formula works for all powers \( n \) except \( n = -1 \), because it would involve division by zero. Instead, for \( n = -1 \), the integral is \( \ln|x| + C \).
When applying the power rule, always increase the exponent by one and divide by the new exponent. For instance, integrating \( x^{-2} \) involves increasing \(-2\) to \(-1\), then dividing by \(-1\), resulting in \( -\frac{1}{x} + C \). This rule simplifies the process of integrating polynomial terms.

Substitution is a handy technique that simplifies the process of integration, particularly when dealing with composite functions. The goal of substitution is to transform a difficult integral into a simpler form that can be easily integrated.
The process involves the following steps:

• Select a substitution \( u = g(x) \).
• Replace \( dx \) with \( du = g'(x)dx \).
• Substitute the variables and integrate with respect to \( u \).
• Finally, substitute back to express the integral in terms of the original variable \( x \).

Let’s take the integral \( \int (x+1)^{-1/3} \, dx \) as an example. By setting \( u = x + 1 \), we transform the integral to \( \int u^{-1/3} \, du \), which is straightforward to solve using the power rule, resulting in \( \frac{3}{2}(x+1)^{2/3} + C \). Substitution simplifies complex integrals by turning them into more manageable ones.

The arcsine function frequently appears when dealing with integrals involving expressions of the form \( \frac{1}{\sqrt{1-x^2}} \). This is because the derivative of \( \arcsin(x) \) is \( \frac{1}{\sqrt{1-x^2}} \), thus naturally arising in integrals that fit this mould.
Key integrals involving the arcsine function include:

• \( \int \frac{1}{\sqrt{1-x^2}} \, dx = \arcsin(x) + C \)
• For more complex expressions: complete the square or use a substitution to transform the integral into this standard form.

For example, consider the integral \( \int \frac{1}{\sqrt{4-x^2}} \, dx \). By recognizing it as \( \int \frac{1}{\sqrt{2^2-x^2}} \, dx \), you can directly apply the arcsine formula, yielding \( \arcsin\left(\frac{x}{2}\right) + C \). This showcases the utility of arcsine in resolving integrals with quadratic expressions within a square root.

The arctangent function plays a crucial role in integrations involving expressions of the form \( \frac{1}{x^2 + a} \). The derivative of \( \arctan(x) \) is \( \frac{1}{1+x^2} \), leading to its presence in indefinite integrals.
Some important arctangent integrals are:

• \( \int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C \)
• With any expression resembling \( \int \frac{1}{x^2 + bx + c} \, dx \), a complete the square approach converts it into the required form.

For instance, when integrating \( \frac{1}{x^2+6x+13} \), complete the square to form \((x+3)^2 + 4\). This matches the arctangent format, resulting in \( \frac{1}{2} \arctan\left(\frac{x+3}{2}\right) + C \). The arctangent function streamlines integration when quadratic expressions are present in the denominator.