91Ó°ÊÓ

In calculus, a definite integral represents the area under a curve between two specific points. When evaluating a definite integral, you find the accumulated value of a function from one point to another. This is different from an indefinite integral, which represents a family of functions and includes a constant of integration.
To compute a definite integral, follow these steps:

• Find the antiderivative (indefinite integral) of the function.
• Evaluate the antiderivative at the upper limit of integration (b).
• Evaluate the antiderivative at the lower limit of integration (a).
• Subtract the value at the lower limit from the value at the upper limit: \[ \text{Definite Integral} = F(b) - F(a) \]

In our example, the integral we evaluated is:
\[ \int_{2}^{6}\frac{3 x + \sqrt{x}}{4 x^{3}} \, dx \]
By calculating this integral, we're finding the total accumulation of the function from x = 2 to x = 6.

The power rule is a basic but crucial tool for integrating functions involving powers of x. It states that for any real number n different from -1:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
This rule can be applied to any term where x is raised to a power. In our problem, we use the power rule to integrate the terms separately:

• For \int x^{-2} \, dx, we get:
• \[ \int x^{-2} \, dx = \frac{x^{-2+1}}{-2+1} = \-x^{-1} = \-\frac{1}{x} \]
• For \int x^{-5/2} \, dx, we get:
• \[ \int x^{-5/2} \, dx = \frac{x^{-5/2+1}}{-5/2+1} = \frac{x^{-3/2}}{-3/2} = \frac{-2}{3} x^{-3/2} = \-\frac{2}{3} \frac{1}{x^{3/2}} \]

Using the power rule makes integration straightforward, even for complex terms.

Integral simplification is the process of transforming a complicated integrand into a simpler form that is easier to integrate. This often includes:

• Splitting a fraction into separate terms.
• Factoring out constants.
• Combining like terms.

For the given problem, we first split the fraction:
\[ \int_{2}^{6} \left( \frac{3x}{4x^{3}} + \frac{\sqrt{x}}{4x^{3}} \right) dx \]
Then, simplify each term:
\[ \frac{3x}{4x^{3}} = \frac{3}{4x^{2}} \]
\[ \frac{\sqrt{x}}{4x^{3}} = \frac{x^{1/2}}{4x^{3}} = \frac{1}{4x^{5/2}} \]
To make integration easier, we factor out the constants and integrate each simplified term separately:
\[ \frac{3}{4} \int_{2}^{6} \frac{1}{x^{2}} \, dx + \frac{1}{4} \int_{2}^{6} \frac{1}{x^{5/2}} \, dx \]
By simplifying the integrand, we've broken down a tough problem into manageable parts, making the integration process much smoother.