91Ó°ÊÓ

Definite integrals allow us to find the total accumulation of a quantity, like area under a curve, over a specified interval. Unlike indefinite integrals, which deal with antiderivatives without bounds, definite integrals provide a specific numerical value. Consider the integral \( \int_{a}^{b} f(x) \, dx \). This expression calculates the signed area between the curve of \( f(x) \) and the x-axis from \( x = a \) to \( x = b \).

The limits \( a \) and \( b \) in the definite integral are crucial:

• \( a \) is the lower limit and \( b \) is the upper limit.
• The definite integral computes the net area, taking into account regions above and below the x-axis.
• It can be computed by finding the antiderivative \( F(x) \) of \( f(x) \), then evaluating \( F(b) - F(a) \).

Using the substitution method in definite integrals requires adjusting the limits as well. When using substitution, change the variable of integration and update the bounds accordingly, ensuring they match the new variable.

The power rule for integration is a straightforward technique that simplifies finding antiderivatives. It states that if \( n eq -1 \), then the integral of \( x^n \) with respect to \( x \) is \( \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.

Here's how you can use the power rule efficiently:

• First, identify the term in the function that fits the form \( x^n \).
• Apply the rule to find the antiderivative, increasing the exponent by one and dividing by the new exponent.
• If dealing with a definite integral, use the antiderivative to evaluate the integral between the specified bounds.
• Always remember to adjust for any coefficients or constants multiplied with \( x^n \).

In our original problem, once substitution transformed the integral to \( \int u^2 \, du \), we applied the power rule to find its antiderivative as \( \frac{u^3}{3} \).

Trigonometric substitution is a method used to simplify integration, particularly when dealing with expressions involving square roots or even powers of sine and cosine. By substituting a trigonometric identity or function, we can transform the problem into a simpler form.

The process typically involves these steps:

• Identify parts of the integrand (the function to be integrated) that suggest a trigonometric substitution, such as products involving \( \sin x \) or \( \cos x \).
• Select the substitution that will simplify the integral, like \( u = \cos x \) in our exercise, turning the trigonometric expression into a polynomial one.
• Calculate \( du \), the derivative of the substitution, to replace the differential \( dx \).
• Re-express the entire integral in terms of the new variable.
• Complete the integration and then substitute back the original trigonometric expressions if necessary.

In the case of our problem, using \( u = \cos x \) reduced the integrand to a simple polynomial, which we successfully integrated using the power rule.