91Ó°ÊÓ

Integration by parts is a technique used to integrate products of functions. When you have an integral of the form \( \int u \, dv \), this method is useful. You can break it down using the formula:

• \( \int u \, dv = uv - \int v \, du \)

Here, you need to choose both \( u \) and \( dv \) cleverly. Typically, follow the LIATE rule for setting \( u \):

• Logarithmic functions
• Inverse trigonometric functions
• Algebraic functions
• Trigonometric functions
• Exponential functions

This approach prioritizes which part of the product to differentiate. However, in the current integral problem \( \int \frac{2x-3}{e^{x^2-3x+1}} \, dx \), no obvious product setup aligns well with this method, indicating substitution is more appropriate.

The substitution method, also known as \( u \)-substitution, simplifies integrals by changing variables. You pick a substitution \( u = g(x) \), typically simplifying the problem:

• Write \( du = g'(x) \, dx \).
• Translate the entire integral into terms of \( u \).

For instance, in the provided integral \( \int \frac{2x-3}{e^{x^2-3x+1}} \, dx \), we observed that the expression \( x^2-3x+1 \) holds potential. Its derivative, \( 2x-3 \), is present in the numerator, suggesting substitution:

• Let \( u = x^2-3x+1 \) and \( du = (2x-3) \, dx \).
• This transforms the integral to \( \int \frac{1}{e^u} \, du \).

This approach makes the integral much easier to solve.

While the focus problem is an indefinite integral, it's worth noting when an integral covers specific bounds, it becomes 'definite'. Definite integration finds the accumulated area under a curve between bounds \( a \) and \( b \):

• \( \int_{a}^{b} f(x) \, dx \)

Evaluate using the Fundamental Theorem of Calculus:

• First, find the antiderivative \( F(x) \).
• Calculate \( F(b) - F(a) \) for the net area.

Definite integrals give a specific numerical result, unlike indefinite integrals that provide a general function plus a constant \( C \). Moreover, substituting can help simplify definite integrals as well, similar to how it aids indefinite integrals.

Antiderivatives, also known as indefinite integrals, are functions whose derivative brings you back to the original function. When integrating \( f(x) \), you seek a function \( F(x) \) such that \( F'(x) = f(x) \).

• General form: \( \int f(x) \, dx = F(x) + C \)

The constant \( C \) represents the family of antiderivatives since any constant differentiates to zero, making multiple solutions possible.
In our particular problem, we simplified the expression via substitution into \( \int \frac{1}{e^u} \, du \), finding the antiderivative \(-e^{-u}\). By substituting back, we arrived at the solution \(-e^{-(x^2 - 3x + 1)} + C\), confirming it's the antiderivative of the given integral.
This process highlights how substitution effectively finds antiderivatives, particularly when direct integration isn't straightforward.