91Ó°ÊÓ

Integration by substitution is a method used to simplify the process of finding an integral, particularly when dealing with complicated functions. Think of it like untying a knot to make things easier. In our problem, we start by identifying a suitable substitution. Let's choose a part of the integrand, like a function or its component, and set it equal to a new variable.

In this exercise, we defined the substitution as:

• Let \( u = 3ax + bx^3 \).

The goal is to transform the integral into an easier form. Calculating the derivative of our substitution helps us connect the new variable back to the original variable. Here, the derivative is:

• \( du = (3a + 3bx^2)dx \).

This derivative shows us how \( dx \) relates to \( du \). By substitution, we simplify the given integral by replacing original parts, making the integral easier to solve.

The power rule of integration is a simple yet powerful tool. It helps solve integrals of functions raised to a power. It says:

• \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \], where \( n eq -1 \).

For our exercise, after making the substitution, we simplified the integral to

• \( \frac{1}{3} \int u^{-1/2} \, du \).

The function \( u^{-1/2} \) fits exactly into the power rule. According to this rule, solving the integral gives:

• \( 2u^{1/2} + C \).

By applying this rule, we easily find the specific form of the integral without complex calculations.

Simplifying an integral involves breaking down complexity into more manageable parts. Let's see how this works step-by-step:

First, use substitution to change variables, identifying parts of the integrand that make integration tough. By setting \( u = 3ax + bx^3 \), we plugged-in and simplified the problem's variables.

Next, replace all variables in the integral as much as possible. In this problem:

• Transform \( (a + bx^2) \) by using the expression found from the derivative.
• Substitute \( dx \) with \( \frac{du}{3a + 3bx^2} \).

This step turned the integral into a simpler form:

• \( \frac{1}{3} \int \frac{du}{\sqrt{u}} \).

Lastly, after integrating, switch back from our new variable \( u \) to the original variable \( x \). Replace \( u \) using our initial substitution to express the solution:

• \( \frac{2}{3} \sqrt{3ax + bx^3} + C \).

Simplifying makes the whole process of integration much more approachable and manageable!