Problem 21 - The Calculus with Analytic Geometry

Question

\(\int \frac{\left(x^{2}+2 x\right) d x}{\sqrt{x^{3}+3 x^{2}+1}}\)

Step-by-Step Solution

Answer

\( \frac{2}{3} \sqrt{x^3 + 3x^2 + 1} + C \)

Step 1 - Identify the substitution

Notice that the integrand involves a square root of a polynomial. Let's use the substitution method. Let’s set \[ u = x^3 + 3x^2 + 1 \]and then find the differential \( du \).

Step 2 - Compute the differential

To find the differential \( du \), differentiate \( u \) with respect to \( x \): \[ \frac{du}{dx} = \frac{d}{dx}(x^3 + 3x^2 + 1) = 3x^2 + 6x \] So, \( du = (3x^2 + 6x) \, dx \).

Step 3 - Rewrite the integrand

Express the integral in terms of \( u \). Notice that:\[ \frac{du}{3} = (x^2 + 2x) \, dx \]So, the integral becomes:\[ \frac{1}{3} \int \frac{du}{\sqrt{u}} \]

Step 4 - Simplify the integrand

Simplify the integrand by rewriting \( 1/\sqrt{u} \) as \( u^{-1/2} \):\[ \frac{1}{3} \int u^{-1/2} du \]

Step 5 - Integrate

Integrate \( u^{-1/2} \): \[ \frac{1}{3} \int u^{-1/2} du = \frac{1}{3} \left(2 \sqrt{u} \right) + C = \frac{2}{3} \sqrt{u} + C \]

Step 6 - Substitute back in terms of \(x\)

Replace \( u \) with its expression in terms of \( x \): \[ \frac{2}{3} \sqrt{x^3 + 3x^2 + 1} + C \]

Key Concepts

Integral Calculus

Integral Calculus involves finding the antiderivative or getting the area under a curve. In this exercise, we work with an indefinite integral, which means that we are looking for the general form of an antiderivative without specific limits.

Our goal is to simplify the integrand, which is the function we're integrating, before finding the antiderivative. Sometimes, this requires special techniques, such as substitution, to make the integral easier to solve. When you identify a complex expression within an integrand, consider simplifying it by substituting it with a new variable. This often transforms the integral into a form that is easier to handle.

Substitution Method

The substitution method, also known as u-substitution, is a powerful technique to simplify integrals.

Here's how it works:

• Identify a part of the integrand to substitute with a new variable, typically denoted as 'u'.
• Find the differential (du) of the new variable in terms of the original variable (x).
• Replace the identified part of the integrand and the differential in terms of 'u'.

In our problem, we start by letting \(u = x^3 + 3x^2 + 1\). Then, find the differential, \(du = (3x^2 + 6x) dx\). This substitution transforms a complicated integrand into a simpler one, allowing us to integrate with ease.

Differentiation

Differentiation is essential in the substitution method.

In our example, to find the differential \(du\), we need to differentiate \(u = x^3 + 3x^2 + 1\) with respect to \(x\):

\[\frac{du}{dx} = \frac{d}{dx}(x^3 + 3x^2 + 1) = 3x^2 + 6x\] \
This gives us \(du = (3x^2 + 6x)dx\).

Differentiation allows us to express the differential in terms of dx, thus connecting the original variable \(x\) to the substituted variable \(u\). Knowing how to differentiate polynomials is a crucial part of applying the substitution method correctly.

Polynomials

Polynomials are expressions consisting of variables and coefficients combined using addition, subtraction, and multiplication. In our integral, we are dealing with polynomials in both the numerator and inside the square root.

Recognizing these polynomial forms is crucial, as they guide us toward appropriate substitution choices. For example, observing that \(x^2 + 2x\) and the expression inside the square root \(x^3 + 3x^2 + 1\) share derivative relationships, nudges us towards a substitution that simplifies the integral.

Understanding how to manipulate and differentiate polynomials makes solving integrals, including those necessitating the substitution method, much more feasible.