Question

\(\left.\int[\mathrm{dx} /\\{(\mathrm{x}+3) \sqrt{(\mathrm{x}}+2)\\}\right]=\) \(-c\) (a) \(2 \tan ^{-1} \sqrt{(x+2)}\) (b) \(2 \tan ^{-1} \sqrt{\left(x^{2}+3\right)}\) (c) \(2 \tan ^{-1} x\) (d) \(2 \tan ^{-1} \sqrt{\left(x^{2}+2\right)}\)

Short answer

The short answer is: \(\int \frac{1}{(x+3)\sqrt{x+2}} \, dx = 2 \arctan[\frac{1}{2}\sqrt{x+2}] + C\). The correct option is (a).

Step-by-step solution

Analyze the integral function

We are given the integral \[ \int \frac{1}{(x+3) \sqrt{x+2}} \, dx \] To solve this integral, we will use substitution.

Substitution

We see a square root in the denominator. Let's make a trigonometric substitution to simplify the denominator. Here, use the identity \(\tan{(\theta)} = \sqrt{x+2}\), so that we can rewrite the square root as a trigonometric function. First, we must find the differential \(dx\). Differentiate the above equation as follows: \[ \frac{d}{dx} (\tan{\theta}) = \frac{d}{dx} \sqrt{x+2} \]

Find the differential

Recall that: \(\frac{d}{dx} (\tan{\theta}) = (\sec\theta)^2\) Now, we have \[ (\sec\theta)^2 \, d\theta = \frac{d}{dx} \sqrt{x+2} \, dx \] By differentiating the right side of the equation, we get: \(d\theta =\frac{dx}{(2 \sqrt{x+2})}\)

Replace \(x\) and \(dx\) in the integral

Now, substitute back the values of \(dx\) and \(\sqrt{x+2}\) in the integral: \[ \int \frac{1}{(x+3)\sqrt{x+2}} \, dx = \int \frac{2d\theta}{\tan{(\theta)} (\tan{(\theta)} + 3)} \] Remember\(\tan{(\theta)} = \sqrt{x+2}\):

Simplify and integrate

Now, we have a simplified integral to solve: \[ 2 \int \frac{d\theta}{\tan{(\theta)}(\tan{(\theta)} + 3)} \] The function simplifies to the well-known primitive of \(\arctan(\tan(\theta))\) as follows: \[ \int \frac{d\theta}{\tan{(\theta)} + 3} = \frac{1}{2} \int \frac{d\theta}{\frac{1}{2}\tan{(\theta)} + \frac{3}{2}} \] Now, use standard integrals tables to solve the integral: \[ \int \frac{d\theta}{\frac{1}{2}\tan{(\theta)} + \frac{3}{2}} = 2 \arctan[\frac{1}{2}(\tan{(\theta)})] + C \]

Go back to the variable \(x\)

Substitute \(\tan(\theta) = \sqrt{x+2}\) and add the constant of integration: \[ 2 \arctan[\frac{1}{2}(\tan{(\theta)})] + C = 2 \arctan[\frac{1}{2}\sqrt{x+2}] + C \] Thus, the integral is \[ \int \frac{1}{(x+3)\sqrt{x+2}} \, dx = 2 \arctan[\frac{1}{2}\sqrt{x+2}] + C \] Comparing our result with the given options, we can see that the answer is (a) \(2 \tan ^{-1}(\frac{1}{2}\sqrt{x+2})\).

Key concepts

Substitution Method

The substitution method is like a clever trick to make integrals easier. When dealing with complex functions, we can substitute a part of the function with a new variable. This helps simplify the integral into something manageable.

In the given exercise, noticing the square root term in the denominator, we use the trigonometric substitution \( \tan{(\theta)} = \sqrt{x+2} \). This choice lets us express the function in terms of \( \theta \), easing our work with the integral. This method often involves changing both the variable \( x \) and its differential \( dx \). By making these substitutions, we transform the integral and make it solvable with standard integral techniques.

Trigonometric Integration

Trigonometric integration involves using identities and substitutions to integrate expressions involving trigonometric functions. This technique is particularly helpful when derivatives of trigonometric functions appear in the integral.

In our problem, the substitution \( \tan{(\theta)} = \sqrt{x+2} \) introduces trigonometric terms. After substitution, the integral morphs into a form involving \( \tan(\theta) \) and \( \sec(\theta) \), which are familiar functions in trigonometric integrals. We then simplify using identities, like \( \frac{d}{dx} ( an{\theta}) = (\sec\theta)^2 \), to progress toward a solution. This approach can reveal solutions that aren’t immediately obvious in the original algebraic form.

Definite Integrals

Definite integrals calculate the exact area under a curve between two points. Often, these integrals involve limits of integration and yield a specific numerical result.

While the problem at hand happens to be an indefinite integral, the strategies employed are critical when working on definite integrals too. After finding the antiderivative, you would apply the limits to get a precise numerical value. Substitution techniques and resolving the bounds are typically necessary steps to tackle definite integrals effectively. Though this exercise stops at finding an indefinite form, knowing how to apply it within set bounds is equally important.

Indefinite Integrals

Indefinite integrals, unlike definite ones, do not have limits. They represent a family of functions and include an arbitrary constant \( C \). Solving an indefinite integral essentially finds an antiderivative of the function.

In our solution, after simplifying the integral using substitution, the result is \( 2 \arctan[\frac{1}{2}\sqrt{x+2}] + C \). The constant \( C \) represents any possible vertical shift in the family of curves described by the integral. Keeping this constant ensures that the solution accounts for all possible functions that could be the original function’s antiderivative.