91Ó°ÊÓ

Definite integrals are one of the core concepts in calculus. They represent the area under the curve of a function between two specified points.
For instance, \[\text{∫_{a}^{b} f(x) \, dx}\] represents the area under the function \(f(x)\) from \(x = a\) to \(x = b\). In contrast to indefinite integrals that include a constant of integration (\(C\)), definite integrals produce a numerical value.
In some cases, transforming a definite integral into a more straightforward form is necessary to simplify calculations. This transformation often involves a technique known as u-substitution.
For example, consider transforming an integral using u-substitution: we can replace a more complex variable with a simpler one.

U-substitution is a powerful technique to simplify integrals, making them easier to solve.
It's particularly useful when dealing with integrals involving a function and its derivative. Here’s a step-by-step process to use u-substitution:

• Identify the substitution: Choose a substitution that simplifies the integral. For instance, in \[\text{∫ x \, sin(x^2) \, dx}\], we let \(u = x^2\).
• Calculate the differential: Find \(du\) in terms of \(dx\). In this case, \(du = 2x \, dx\), or rearranged, \( \frac{1}{2} du = x \, dx\).
• Rewrite the integral: Substitute \(u\) and \(du\) into the integral. This turns \[ \text{∫ x \, sin(x^2) \, dx} = \text{∫ \frac{1}{2} \, sin(u) \, du}\]
• Integrate with respect to \(u\): Solve the simpler integral:
\[ \text{∫ \frac{1}{2} \, sin(u) \, du} = \frac{1}{2} (-\text{cos(u)}) + C = -\frac{1}{2} \text{cos(u)} + C \]

Remember to substitute back \(u = x^2\) once you integrate:
\[ -\frac{1}{2} \text{cos}(x^2) + C \] Finally, always check your answer by differentiating back to see if you then retrieve the original integrand.

The chain rule is an essential tool in calculus, particularly in differentiation. It allows us to differentiate composite functions.
If you have a function \(y = f(g(x))\), the chain rule states: \[ \frac{dy}{dx} = f'(g(x)) \times g'(x) \]
Let’s use the chain rule to verify our integral solution. After substituting back in our example, we have \[ -\frac{1}{2} \text{cos}(x^2) + C \]
To differentiate, apply the chain rule: \[ \frac{d}{dx} \big(-\frac{1}{2} \text{cos}(x^2)\big) = -\frac{1}{2} \times \text{sin}(x^2) \times 2x \]
Simplifying, we get: \[ -\frac{1}{2} \times 2x \times \text{sin}(x^2) = x \times \text{sin}(x^2) \]
This matches the original integrand. Hence, using the chain rule confirms the correctness of our substitution and integration process.