Problem 65 - Calculus: Early Transcendentals

Question

Evaluate the definite integral. \( \displaystyle \int^a_0 x\sqrt{x^2 + a^2} \, dx \) \( (a > 0) \)

Step-by-Step Solution

Answer

The evaluated definite integral is \( \frac{a^3}{3} \left( 1 - \frac{1}{2 \sqrt{2}} \right) \).

Step 1: Identify the Problem Type

We need to evaluate the definite integral \( \int^a_0 x\sqrt{x^2 + a^2} \, dx \). This is a definite integral involving a product of a polynomial and a square root function.

Step 2: Choose a Suitable Technique

Observing the integral \( \int x\sqrt{x^2 + a^2} \, dx \), we can use the substitution method. Let us try a trigonometric substitution to simplify the square root term.

Step 3: Make the Substitution

Use the substitution \( x = a \tan(\theta) \), which implies \( dx = a \sec^2(\theta) \, d\theta \). Then \( x^2 + a^2 = a^2 \sec^2(\theta) \). The limits change from \( x=0 \) to \( x=a \). For \( x=0 \), \( \theta=0 \); and for \( x=a \), \( \theta=\frac{\pi}{4} \).

Step 4: Substitute and Simplify the Integral

Substituting into the integral, we have:\[\int_0^{\frac{\pi}{4}} a \tan(\theta) \cdot a \sec(\theta) \cdot a \sec^2(\theta) \, d\theta = a^3 \int_0^{\frac{\pi}{4}} \tan(\theta) \sec^3(\theta) \, d\theta.\]

Step 5: Integrate Using a Trigonometric Identity

Using the identity \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) and \( \sec(\theta) = \frac{1}{\cos(\theta)} \), rewrite the integral as:\[a^3 \int_0^{\frac{\pi}{4}} \frac{\sin(\theta)}{\cos^4(\theta)} \, d\theta.\]This can be integrated using a trigonometric substitution or a reduction formula.

Step 6: Solve the Integral

To solve \( \int \frac{\sin(\theta)}{\cos^4(\theta)} \, d\theta \), consider a substitution \( u = \cos(\theta) \), with \( du = -\sin(\theta) \, d\theta \). The integral becomes:\[a^3 \left[-\int_1^{\frac{1}{\sqrt{2}}} \frac{1}{u^4} \, du \right] = a^3 \left[\frac{1}{3}\left(u^{-3}\right)\right]_1^{\frac{1}{\sqrt{2}}} = a^3 \left[ \frac{1}{3} \left(\frac{1}{2\sqrt{2}} - 1 \right) \right].\]

Step 7: Evaluate and Simplify

Evaluate the expression:\[a^3 \cdot \frac{1}{3} \left( 1 - \frac{1}{2 \sqrt{2}} \right) = \frac{a^3}{3} \left( 1 - \frac{1}{2 \sqrt{2}} \right).\]

Step 8: Return to Original Variables

There are no additional variables since \( \theta \) is fully integrated based on trigonometric limits.

Key Concepts

Trigonometric Substitution

Trigonometric substitution is an excellent method to tackle integrals with expressions like \( \sqrt{x^2 + a^2} \). This method relies on transforming the variable to a trigonometric function that simplifies the radical.

In our case, by substituting \( x = a \tan(\theta) \), the square root simplifies to \( a \sec(\theta) \). This reduces complexity by leveraging trigonometric identities, making calculus more straightforward.

• This substitution is valid because \( \tan^2(\theta) + 1 = \sec^2(\theta) \).
• It transforms the square root into a function solely in terms of \( \theta \), allowing easier integration.

The limits of integration change accordingly — when \( x = 0 \), \( \theta = 0 \), and for \( x = a \), \( \theta = \frac{\pi}{4} \). This substitution is crucial in handling integrals where direct methods fall short.

This process involves seeing patterns in polynomial and square root functions and strategically applying trigonometric identities.

Integration Techniques

Integrating complex expressions often requires a combination of techniques to simplify and evaluate the definite integral efficiently.

Initially, we used trigonometric substitution to convert the integral into a trigonometric form. The factor of \( a^3 \) highlights the effect of substitution on coefficients. This results in reducing the integral to a form that's more accessible using trigonometric identities.

Subsequently, identities such as \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) and \( \sec(\theta) = \frac{1}{\cos(\theta)} \) are used to express the integral in terms of \( \sin(\theta) \) and \( \cos(\theta) \).

• Additionally, a further substitution, \( u = \cos(\theta) \), simplifies the integral into a polynomial form \( \frac{1}{u^4} \) in terms of \( u \).
• This method allows us to use basic integration rules of polynomials to solve for \( u \).

The expertise in applying the correct technique lies in recognizing when and how to combine various methods like substitution, trigonometric identities, and algebraic manipulation for successful integration.

Polynomial and Square Root Function Integration

The original function \( x\sqrt{x^2 + a^2} \) combines polynomial and square root elements, necessitating a delicate approach to integration. By recognizing its components—a linear polynomial (\( x \)) and a square root of a square plus constant (\( \sqrt{x^2 + a^2} \))—we can deploy trigonometric substitution which precisely deals with such hybrid forms.

The integral transformation converts these parts into trigonometric functions, which have well-documented identities useful for integration.

• We express the original product in terms of \( \theta \), resulting in a manageable trigonometric integral \( a^3 \int \tan(\theta)\sec^3(\theta) \, d\theta \).
• This integration path is logical as both the difficulty of directly integrating \( x \sqrt{x^2 + a^2} \), and the application of trigonometric identities reveal a straightforward equivalent formulation in terms of \( \theta \).

Ultimately, substituting back and evaluating under the defined limits yields the integral's solution. Returning to the context of \( x \) completes the process without needing further transformations. By blending polynomial and radical manipulation with trigonometry, complex integrals become navigable.