Problem 43 - Calculus Single Variable

Question

Each of the integrands involves an expression of the form \(a^{2}-b^{2} x^{2}, a^{2}+b^{2} x^{2},\) or \(b^{2} x^{2}-a^{2} .\) Use an indirect substitution of the form \(x=(a / b) \sin (\theta), x=(a / b)\) \(\tan (\theta),\) or \(x=(a / b) \sec (\theta)\) to calculate the given integral. $$ \int_{0}^{2} \frac{6}{\left(9+4 x^{2}\right)^{3 / 2}} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{4}{15} \).

Step 1: Analyze the Integral

The given integral is \( \int_{0}^{2} \frac{6}{(9 + 4x^2)^{3/2}} \, dx \). This resembles the form \( a^2 + b^2 x^2 \) where \( a^2 = 9 \) and \( b^2 = 4 \), thus \( a = 3 \) and \( b = 2 \). This suggests using substitution \( x = (a/b) \tan(\theta) \).

Step 2: Substitute \(x = \frac{3}{2}\tan\theta \)

Set \( x = \frac{3}{2} \tan \theta \). Then, \( dx = \frac{3}{2} \sec^2 \theta \, d\theta \). Substitute in the integral: \[ \int \frac{6}{\left( 9 + 4 \left( \frac{9}{4} \tan^2 \theta \right) \right)^{3/2}} \cdot \frac{3}{2} \sec^2 \theta \, d\theta. \] Simplify to \( \int \frac{9 \sec^2 \theta}{9^{3/2} \sec^3 \theta} \, d\theta \).

Step 3: Simplify the Integrand

Since \( 9 + 4x^2 = 9 \sec^2 \theta \), the substitution simplifies the expression under the radical to \( 9 \sec^2 \theta \). Hence the expression \( (9 \sec^2 \theta)^{3/2} = 27 \sec^3 \theta \). The integrand becomes \( \frac{9 \sec^2 \theta}{27 \sec^3 \theta} = \frac{1}{3\sec \theta} = \frac{1}{3} \cos \theta \).

Step 4: Integrate \( \frac{1}{3} \cos \theta \)

The integral now is \( \int \frac{1}{3} \cos \theta \, d\theta = \frac{1}{3} \int \cos \theta \, d\theta = \frac{1}{3} \sin \theta + C \).

Step 5: Back-substitute \( \theta \) in terms of \( x \)

Since \( x = \frac{3}{2} \tan \theta \), \( \tan \theta = \frac{2x}{3} \). Therefore from \( \tan \theta = \frac{2x}{3} \), we find \( \sin \theta = \frac{\frac{2x}{3}}{\sqrt{1 + \left( \frac{2x}{3} \right)^2}} = \frac{2x}{\sqrt{9 + 4x^2}} \). Substitute back: \( \frac{1}{3} \sin \theta = \frac{2x}{3\sqrt{9 + 4x^2}} \).

Step 6: Evaluate the Definite Integral

Now evaluate \( \left[ \frac{2x}{3\sqrt{9 + 4x^2}} \right]_{0}^{2} \). For \( x = 2 \), \( \frac{2 \cdot 2}{3 \cdot \sqrt{9 + 4 \cdot 2^2}} = \frac{4}{3 \cdot 5} = \frac{4}{15} \). For \( x = 0 \), it evaluates to \( 0 \). The final result is \( \frac{4}{15} - 0 = \frac{4}{15} \).

Key Concepts

Integrals with Trigonometric Substitution

When tackling integrals involving expressions like \( a^2 + b^2 x^2 \), trigonometric substitution is a powerful technique. This involves replacing parts of the integral with trigonometric functions to simplify the expression. This is particularly useful because trigonometric identities can transform complicated algebraic expressions into simpler ones.

In the provided exercise, the integral \( \int_{0}^{2} \frac{6}{(9 + 4x^2)^{3/2}} \, dx \) fits the prototype \( a^2 + b^2 x^2 \) with \( a = 3 \) and \( b = 2 \). We use the substitution \( x = (a/b) \tan(\theta) \) because it matches the form most closely related to the secant identity, \( \sec^2 \theta = 1 + \tan^2 \theta \).

After substituting, we transform the integral, making it easier to solve by reducing powers of secant, simplifying our calculations. This approach efficiently shifts the problem from a formidable algebraic form to something more comfortable to integrate, capitalizing on the properties of trigonometric functions.

Indirect Substitution in Integrals

Indirect substitution in integrals is a process where an intermediate variable, such as \( \theta \) in trigonometric substitution, is introduced. This method reshapes the integral into a more workable expression. Instead of directly substituting \( x \) back, the process relies on derivative and trigonometric identities to progress.

In our case, substituting \( x = \frac{3}{2} \tan(\theta) \) led to expressing \( dx \) in terms of \( d\theta \) as \( dx = \frac{3}{2} \sec^2 \theta \, d\theta \). This transformation leverages the relationship between tangent and secant given by \( \sec^2 \theta = 1 + \tan^2 \theta \).

The indirect substitution simplifies the integration process by reducing the original polynomial under the radical to a form that is much easier to integrate using basic trigonometric functions. It’s a terrific strategy for handling integrals that otherwise would be tricky due to complex algebraic expressions.

Calculus Step by Step Solution

Solving calculus problems with a step-by-step approach is crucial for understanding complex concepts like integrals. Step-by-step solution involves breaking down the problem into more straightforward subtasks. Not only does this make it easier to follow, but it also ensures that you understand each stage of the solution process.

For the given integral \( \int_{0}^{2} \frac{6}{(9 + 4x^2)^{3/2}} \, dx \), the step-by-step solution began with analyzing the integral and identifying an appropriate substitution. Recognizing that \( a^2 + b^2 x^2 \) is present, the substitution \( x = \frac{3}{2} \tan(\theta) \) was used to simplify the integral significantly.

Then, the integral was further reduced by substituting \( dx = \frac{3}{2} \sec^2 \theta \, d\theta \), and eventually integrating \( \frac{1}{3} \cos(\theta) \, d\theta \). Finally, back-substitution was performed to express the result in terms of \( x \), and the definite integral was evaluated over the specified limits. Each step focuses on transforming the problem into simpler parts, ensuring accuracy and clarity throughout the procedure.