Problem 7 - Applied Calculus

Question

Find the integrals .Check your answers by differentiation. $$\int \frac{x}{\sqrt{x^{2}+4}} d x$$

Step-by-Step Solution

Answer

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

Step 1: Identify a Suitable Substitution

The integral \( \int \frac{x}{\sqrt{x^{2}+4}} \, dx \) can be solved using a trigonometric substitution. We notice that the expression under the square root suggests using the substitution \( x = 2\tan(\theta) \), which transforms \( x^2 + 4 \) into a form that can involve trigonometric identities.

Step 2: Perform the Trigonometric Substitution

Using the substitution \( x = 2\tan(\theta) \), we have \( dx = 2\sec^2(\theta) \, d\theta \). Substituting these into the integral gives: \[\int \frac{2\tan(\theta)}{\sqrt{4\tan^2(\theta) + 4}} \cdot 2\sec^2(\theta) \, d\theta\] The square root simplifies using \( 1 + \tan^2(\theta) = \sec^2(\theta) \), leading to the expression: \[ 4 \int \tan(\theta) \cdot \sec(\theta) \, d\theta \] Which simplifies to:\[ 4 \int \sin(\theta) \, d\theta \]

Step 3: Integrate the Simplified Expression

The integral \( \int \sin(\theta) \, d\theta \) can be directly integrated to obtain: \[ -\cos(\theta) + C \] Where \( C \) is the constant of integration. Therefore, the expression becomes: \[ -4\cos(\theta) + C \]

Step 4: Back Substitute for \( \theta \)

Recall that \( x = 2\tan(\theta) \). Hence, \( \tan(\theta) = \frac{x}{2} \) and \( \cos(\theta) = \frac{2}{\sqrt{x^2+4}} \) based on the identity \( \tan^2(\theta) + 1 = \sec^2(\theta) \). Substituting back gives: \[ -4 \cdot \frac{2}{\sqrt{x^2+4}} + C = -\frac{8}{\sqrt{x^2+4}} + C \]

Step 5: Check by Differentiation

To verify, differentiate the result \( -\frac{8}{\sqrt{x^2+4}} + C \). Applying the chain rule, we find: \[ \frac{d}{dx} \left( -\frac{8}{\sqrt{x^2+4}} \right) = \frac{8x}{(x^2+4)^{3/2}} \] which simplifies algebraically to match the original function \( \frac{x}{\sqrt{x^2+4}} \), confirming our solution.

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a powerful technique used to simplify and solve integrals, especially when dealing with expressions involving square roots. In this exercise, we have an integral with a square root in the denominator:

• \( \int \frac{x}{\sqrt{x^{2}+4}} \, dx \)

The presence of \( x^2 + 4 \) under a square root suggests using trigonometric identities. Here, we use the substitution \( x = 2\tan(\theta) \), as it helps transform the integral into a more manageable form.
With \( x = 2\tan(\theta) \), we find that \( dx = 2\sec^2(\theta) \, d\theta \), simplifying the expression under the square root:

• \( \sqrt{x^2 + 4} = \sqrt{4\tan^2(\theta) + 4} = 2\sec(\theta) \)

The integral transforms based on this substitution, paving the way to apply trigonometric identities like \( 1 + \tan^2(\theta) = \sec^2(\theta) \), which simplifies our work greatly.

Definite Integrals

Although this problem focuses on indefinite integrals, one might wonder how definite integrals come into play with trigonometric substitution. A key aspect is the transformation of limits when substituting variables. Suppose we are solving a definite integral with limits \( a \) to \( b \):

• First, replace the original limits with new limits based on your substitution.

For example, if \( x = 2\tan(\theta) \), we would need to convert both \( a \) and \( b \) into their corresponding \( \theta \) values through \( \theta = \arctan\left(\frac{x}{2}\right) \).
The key takeaway is that integration bounds must also be transformed alongside the variables, ensuring we accurately evaluate the integral.
This ensures a smooth transition and maintains the integrity of the integral within the given interval.

Differentiation Verification

Checking your integration results through differentiation is a crucial step. It confirms the accuracy of your solution.This problem ends with a differentiation step to validate:

• \( -\frac{8}{\sqrt{x^2+4}} + C \)

Use the chain rule to differentiate. When differentiating \( -\frac{8}{\sqrt{x^2+4}} \), apply:

• Derivative of the outer function: \(-8\) times the derivative of \( \frac{1}{\sqrt{x^2+4}} \)
• Chain rule for \( \frac{1}{\sqrt{x^2+4}} \) gives: \( \frac{-x}{(x^2+4)^{3/2}} \)

Finally, combining these components, you recover the original integrand, \( \frac{x}{\sqrt{x^2+4}} \), verifying that your integration was performed correctly.
This method ensures the solution is sound and reinforces the role of differentiation as a powerful tool for verification in calculus.