Problem 32 - Calculus Single Variable

Question

Evaluate the given integral. $$ \int \frac{x^{2}+x}{\sqrt{1-x^{2} / 4}} d x $$

Step-by-Step Solution

Answer

\( 4 \arcsin\left( \frac{x}{2} \right) - 2x\sqrt{1-\frac{x^2}{4}} - 4\sqrt{1-\frac{x^2}{4}} + C \)

Step 1: Simplify the Integral

Recognize that the integral can be simplified by tackling the expression inside. The given integral is \( \int \frac{x^{2}+x}{\sqrt{1-x^{2} / 4}} \, dx \). To simplify, consider rewriting the denominator and see if it hints towards a substitution.

Step 2: Substitution

Apply a trigonometric substitution: Let \( x = 2\sin\theta \). Thus, \( dx = 2\cos\theta\, d\theta \) and \( \sqrt{1 - \frac{x^2}{4}} = \sqrt{1 - \sin^2\theta} = \cos\theta \). Substituting these into the integral gives:\[ \int \frac{(2 \sin\theta)^2 + 2 \sin\theta}{\cos\theta} \cdot 2\cos\theta\, d\theta \].

Step 3: Simplify and Expand

Now simplify the expression: \[ \int (4\sin^2\theta + 2\sin\theta) \cdot 2\, d\theta = \int (8\sin^2\theta + 4\sin\theta) \, d\theta \].

Step 4: Solve Each Integral Term Separately

The expression splits into:\[ \int 8\sin^2\theta\, d\theta + \int 4\sin\theta \, d\theta \].The first part uses a different identity while the second is straightforward.

Step 5: Integrate Using Identities

For \( \int 8\sin^2\theta\, d\theta\), use the identity \( \sin^2\theta = \frac{1 - \cos(2\theta)}{2} \). Substituting gives:\[ \int 8 \left( \frac{1 - \cos(2\theta)}{2} \right) \, d\theta = \int (4 - 4\cos(2\theta)) \, d\theta \].For the second integral, \( \int 4\sin\theta \, d\theta \), directly integrate to get \(-4\cos\theta + C_1\).

Step 6: Calculate and Combine Integrals

The integral \( \int (4 - 4\cos(2\theta)) \, d\theta \) gives:- \( 4\theta \) for the first term.- \( -2\sin(2\theta)\) for the second (using substitution \( \int \cos(2\theta) = \frac{1}{2}\sin(2\theta)\)).Combine everything to have:\[ 4\theta - 2\sin(2\theta) - 4\cos\theta + C \].

Step 7: Back Substitution

Revert back to the original variable \( x \) using \( x = 2\sin\theta \) and \( \theta = \arcsin\left(\frac{x}{2}\right) \). This means:- \( \sin\theta = \frac{x}{2} \), so \( \cos\theta = \sqrt{1-\frac{x^2}{4}} \).- \( \theta = \arcsin\left(\frac{x}{2}\right) \) and- \( \sin(2\theta) = 2\sin\theta \cos\theta = x\sqrt{1-\frac{x^2}{4}} \).

Step 8: Final Expression in terms of x

Substitute back to express everything in terms of \( x \):- \( 4\theta = 4 \arcsin\left( \frac{x}{2} \right) \).- \( -2\sin(2\theta) = -2x\sqrt{1-\frac{x^2}{4}} \).- \( -4\cos\theta = -4\sqrt{1-\frac{x^2}{4}} \).Thus, the solution to the integral is:\[ 4 \arcsin\left( \frac{x}{2} \right) - 2x\sqrt{1-\frac{x^2}{4}} - 4\sqrt{1-\frac{x^2}{4}} + C \].

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a clever technique used in integral calculus to simplify certain integrals. It's especially helpful when dealing with expressions involving square roots, like \( \sqrt{1 - x^2} \). This technique leverages trigonometric identities to transform a complex problem into a more manageable one. Typically, substitutions such as \( x = \sin\theta \), \( x = \cos\theta \), or \( x = \tan\theta \) are used. These correspond to the identities:

• \( \sin^2\theta + \cos^2\theta = 1 \)
• \( 1 + \tan^2\theta = \sec^2\theta \)

In our specific exercise, by setting \( x = 2\sin\theta \), we changed the problem from one involving \( x \) to one involving \( \theta \), which simplifies the integral. This is because the trigonometric identity \( \cos^2\theta = 1 - \sin^2\theta \) allows us to replace \( \sqrt{1 - \sin^2\theta} \) with \( \cos\theta \). This substitution transforms the integral into a simpler form that is easier to evaluate.

Integration Techniques

Integration techniques are a set of methods used to find integrals, especially when direct integration seems challenging. In calculus, various techniques such as substitution, integration by parts, and partial fractions are tools that help solve integrals that do not have simple antiderivatives. In this exercise, after applying trigonometric substitution, the integral became: \[ \int (8\sin^2\theta + 4\sin\theta) \, d\theta \]
This required breaking it down further using another technique - **trigonometric identities**. Using \( \sin^2\theta = \frac{1 - \cos(2\theta)}{2} \), the integral of \( 8\sin^2\theta \) was simplified. This step-by-step transformation helps deal with integrals term by term, ultimately leading to a problem that is manageable within the framework of basic calculus techniques.

Definite and Indefinite Integrals

In integral calculus, understanding the difference between definite and indefinite integrals is crucial. An **indefinite integral** represents a family of functions and includes a constant of integration \( C \), indicating that there are multiple antiderivatives. It looks like this: \( \int f(x)\, dx = F(x) + C \).
On the other hand, a **definite integral** calculates the area under the curve between two points, which results in a numerical value.
In our example, we've dealt with an indefinite integral of the form: \[ \int \frac{x^2 + x}{\sqrt{1 - x^2 / 4}} \, dx \] After transformation and integration processes, we ended with a solution that included \( + C \), signifying an indefinite integral. This indicates that any potential solution could vary by a constant, reflecting the nature of antiderivatives in calculus. Recognizing which type of integral you're solving is crucial, as they require different approaches and interpretations in their results.