Question

For the following exercises, evaluate the integral using the specified method.\(\int \frac{x^{5}}{\left(4 x^{2}+4\right)^{5 / 2}} d x\) using trigonometric substitution

Short answer

The integral is \( \frac{x^6}{6(x^2+2)^3} - \frac{x^8}{8(x^2+2)^4} + C \).

Step-by-step solution

Choose a Trigonometric Substitution

For integrals involving expressions like \( x^2 + a^2 \), trigonometric substitution is useful. Here, set \( x = 2\tan(\theta) \), so \( dx = 2\sec^2(\theta) \ d\theta \). This substitution is chosen because \( 4x^2 + 4 = 4(\tan^2(\theta) + 1) = 4\sec^2(\theta) \).

Simplify the Integral

Substitute \( x = 2\tan(\theta) \) into the integral:\[ \int \frac{(2\tan(\theta))^5}{(4\sec^2(\theta))^{5/2}} \cdot 2\sec^2(\theta) \ d\theta \]Simplify:\[ \int \frac{32\tan^5(\theta) \cdot \sec^2(\theta)}{32\sec^5(\theta)} \ d\theta = \int \tan^5(\theta) \sec^{-3}(\theta) \ d\theta \]

Simplify Further Using Trigonometric Identities

Express \( \tan^5(\theta) \sec^{-3}(\theta) \) using identities:\[ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}, \quad \sec(\theta) = \frac{1}{\cos(\theta)} \]So:\[ \tan^5(\theta) \sec^{-3}(\theta) = \frac{\sin^5(\theta)}{\cos^2(\theta)} \]Express this as \( \sin^5(\theta)(1 - \sin^2(\theta)) \).

Perform the Integration

Let \( u = \sin(\theta) \), then \( du = \cos(\theta) \ d\theta \). Substitute into the integral:\[ \int \sin^5(\theta)(1 - \sin^2(\theta)) \cos(\theta) \ d\theta \rightarrow \int u^5(1-u^2) \ du \]Multiply out and integrate each term:\[ \int (u^5 - u^7) \ du = \frac{u^6}{6} - \frac{u^8}{8} + C \]

Back-substitute to Original Variables

Convert back to \( \theta \) using \( u = \sin(\theta) \):\[ \frac{\sin^6(\theta)}{6} - \frac{\sin^8(\theta)}{8} + C \]Replace \( \sin(\theta) = \frac{x}{\sqrt{x^2 + 2}} \) (from the original substitution) to get the expression back in terms of \( x \).

Present the Final Answer

The final integral in terms of \( x \) is:\[ \frac{\left(\frac{x}{\sqrt{x^2 + 2}}\right)^6}{6} - \frac{\left(\frac{x}{\sqrt{x^2 + 2}}\right)^8}{8} + C \]This simplifies to:\[ \frac{x^6}{6(x^2+2)^3} - \frac{x^8}{8(x^2+2)^4} + C \]

Key concepts

Integration Techniques

Integration techniques are strategies to find the integral of a function. They are essential tools in calculus, enabling us to solve complex problems by transforming them into simpler forms.
One such method is trigonometric substitution, which is especially useful for integrals involving expressions like \( x^2 + a^2 \). By choosing appropriate substitutions, we can turn complicated expressions into trigonometric identities, which are easier to manage.
In our exercise, the substitution \( x = 2\tan(\theta) \) was employed. This choice is convenient because it simplifies the expression \( 4x^2 + 4 \) to \( 4\sec^2(\theta) \), a simple trigonometric function. After substitution, the integral becomes a function of \( \theta \), a variable we can more easily integrate.

• Choose an appropriate substitution based on the form of the integrand.
• Transform the variable and simplify the integral.
• Integrate the simpler function.
• Convert back to original variables if necessary.

Understanding these steps is crucial in applying integration techniques effectively.

Trigonometric Identities

Trigonometric identities are equations that hold true for all values of the involved angles. They are crucial in simplifying integrals involving trigonometric substitutions.
In the exercise, identities like \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \) and \( \sec(\theta) = \frac{1}{\cos(\theta)} \) were used. These helped transform the expression \( \tan^5(\theta) \sec^{-3}(\theta) \) into a form amenable to integration, specifically \( \sin^5(\theta)(1 - \sin^2(\theta)) \).

• Use identities to simplify expressions post substitution.
• Ensure your identities are applied consistently to reduce complexity.
• Convert trigonometric functions into simpler terms, facilitating easier integration.

Mastering trigonometric identities allows for the elegant simplification and calculation of otherwise intimidating integrals.

Calculus Problem Solving

In calculus problem-solving, trigonometric substitutions are often used to solve complex integrals. These problems test your ability to manipulate and evaluate intricate expressions.
Here, selecting \( x = 2\tan(\theta) \) was crucial for converting the original integral into a more straightforward form. Following a series of substitutions and simplifications, complex terms become easier to integrate, showcasing the power of calculus methods.
To tackle calculus problems effectively, follow these guidelines:

• Break down the problem into manageable steps.
• Apply logical substitutions and transformations.
• Use relationships between functions to simplify.
• Iterate through potential solutions and verify reactions back to original variables.

Problem-solving in calculus involves both analytical skills and creativity, making it a rewarding area of study.

Integral Evaluation

Integral evaluation involves finding the indefinite integral of a function, which represents the family of all antiderivatives. This process can sometimes seem daunting but follows a logical progression.
Once we've simplified using trigonometric substitution and identities, the next step is to integrate the resulting function of \( \theta \), with respect to \( d\theta \). For our problem, this process involved converting \( u = \sin(\theta) \) to transform the integral into terms of \( u \).

• Evaluate the integral of the simplified expression.
• Remember the constant of integration, \( C \), which represents an unknown constant.
• Back-substitute to the original variables once integration is complete to show the function in its original variable.
• Simplify further for a clearer final expression.

Integral evaluation provides insights into the behavior of functions, and mastering this skill is pivotal in advanced calculus.