Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

Evaluate each improper integral or show that it diverges. $$ \int_{c}^{2 c} \frac{x d x}{\sqrt{x^{2}+x c-2 c^{2}}}, c>0 $$

Step-by-Step Solution

Answer

The integral evaluates to \(c\), thus it converges.

Step 1: Substitution

Let's use the substitution method to evaluate the integral. We will use the substitution \(u = x^2 + xc - 2c^2\). First, calculate the derivatives: \[\frac{du}{dx} = 2x + c \rightarrow dx = \frac{du}{2x + c}\].The limits transform as:- When \(x = c\), \(u = c^2 + c^2 - 2c^2 = 0\).- When \(x = 2c\), \(u = (2c)^2 + 2c^2 - 2c^2 = 4c^2\).

Step 2: Transform the Integral

Transform the integral using the substitution \(u = x^2 + xc - 2c^2\):\[\int_{0}^{4c^2} \frac{x}{\sqrt{u}} \cdot \frac{du}{2x + c}.\]The \(x\) in numerator cancels with one part of the denominator, simplifying to:\[\frac{1}{2} \int_{0}^{4c^2} \frac{du}{\sqrt{u}}.\]

Step 3: Evaluate the Integral

The integral simplifies to \(\frac{1}{2} \int_{0}^{4c^2} u^{-1/2} du \). Recalling the integral formula for \(u^n\), we find:\[\frac{1}{2} \cdot 2u^{1/2} \Bigg|_0^{4c^2}.\]This evaluates to:\[u_{0}^{4c^2} = \sqrt{4c^2} - \sqrt{0} = 2c.\]

Step 4: Final Evaluation

After evaluating the integral, multiply the result by the constant \(\frac{1}{2}\), giving:\[\frac{1}{2} \cdot 2c = c.\]

Step 5: Conclusion

The improper integral evaluates to \(c\) and does not diverge.

Key Concepts

Substitution Method

The substitution method is a powerful technique used in calculus to simplify complex integrals by changing variables. This technique often transforms an integral into a more manageable form.
In our original exercise, we tackled the integral by using the substitution \( u = x^2 + xc - 2c^2 \). By introducing this new variable \( u \), we can transform both the integral and its limits of integration into a form that is easier to solve.
The derivative of \( u \) with respect to \( x \) is \( \frac{du}{dx} = 2x + c \), so \( dx = \frac{du}{2x + c} \). This substitution transforms the integral, allowing us to simplify the expression before integration. Remember, the goal of substitution is to make the integral easier to evaluate.

Integral Convergence

When dealing with improper integrals, an important consideration is whether the integral converges or diverges. An improper integral is convergent if it results in a finite value as the limits of integration extend to infinity or as they approach an asymptote.
In the exercise we've explored, we are checking for convergence in the integral \( \int_{c}^{2c} \frac{x \, dx}{\sqrt{x^2 + xc - 2c^2}} \). By using substitution and proper transformation, we determine whether the integral reaches a finite value. This particular integral, when evaluated, results in a finite value, indicating convergence.
Convergence is vital as it tells us that the integral is well-defined and meaningful in real-world applications.

Integral Evaluation

Integral evaluation is the process of calculating the value of a definite integral. After substitution, our integral became \( \frac{1}{2} \int_{0}^{4c^2} u^{-1/2} \, du \). To evaluate this, we use the formula for integrating power functions, \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.
Applying it to \( u^{-1/2} \), we find the antiderivative to be \( 2u^{1/2} \). We then evaluate this antiderivative from 0 to \( 4c^2 \) to find the definite value. The solution comes out as \( 2c \).
Integral evaluation requires careful setup and simplification using substitution, making the integration process itself much smoother.

Limits of Integration

The limits of integration define the range over which an integral is evaluated. For our improper integral, the original limits were \( c \) to \( 2c \). After substitution, these limits transformed to 0 and \( 4c^2 \).
Transforming the limits is crucial when using substitution because they remain consistent with the new variable. Proper conversion ensures that the new integral truly represents the original function over its intended domain.
For this exercise, when \( x = c \), \( u = 0 \), and when \( x = 2c \), \( u = 4c^2 \). Recognizing and correctly applying these new limits is essential for evaluating the definite integral accurately.