Problem 38 - Calculus Early Transcendentals

Question

Evaluate each definite integral. \(\int_{0}^{4} \frac{\operatorname{sech}^{2} \sqrt{x}}{\sqrt{x}} d x\)

Step-by-Step Solution

Answer

Question: Evaluate the definite integral \(\int_{0}^{4} \frac{\operatorname{sech}^{2} \sqrt{x}}{\sqrt{x}} \, dx\). Answer: \(\frac{1}{2} \operatorname{tanh}(2)\)

Step 1: Perform substitution

Let's substitute \(u = \sqrt{x}\) to simplify the integral. This means our integrand will be \(\operatorname{sech}^{2}(u)\). We now need to find \(du\) and change the bounds of integration according to our substitution.

Step 2: Calculate the derivative of x with respect to u

To find \(du\), denote \(x = u^2\) and differentiate with respect to \(u\): \(du = 2 u \, dx\). Now, find \(dx\) by dividing both sides by \(2u\): \[ dx= \frac{du}{2u} \]

Step 3: Transform the integrand and bounds of integration

Now substitute \(dx\) and \(u\) into the original integral: \[ \int_{0}^{4} \frac{\operatorname{sech}^{2} \sqrt{x}}{\sqrt{x}} \, dx = \int_{0}^{2} \frac{\operatorname{sech}^{2} u}{u} \cdot \frac{du}{2u}\] The bounds of integration have changed: For the lower bound, when \(x = 0\), \(u = \sqrt{0} = 0\); For the upper bound, when \(x = 4\), \(u = \sqrt{4} = 2\).

Step 4: Simplify the integral

Cancel out the common terms and simplify the integral: \[ \int_{0}^{2} \frac{\operatorname{sech}^{2} u}{u} \cdot \frac{du}{2u} = \int_{0}^{2} \frac{1}{2} \operatorname{sech}^{2} u \, du\]

Step 5: Integrate with respect to u

Now we can integrate the simplified integral, which can be evaluated as: \[ \int_{0}^{2} \frac{1}{2} \operatorname{sech}^{2} u \, du = \frac{1}{2} \int_{0}^{2} \operatorname{sech}^{2} u \, du\] The integral of \(\operatorname{sech}^{2} u\) is \(\operatorname{tanh}(u)\). Thus, the integral becomes: \[ \frac{1}{2} \int_{0}^{2} \operatorname{sech}^{2} u \, du = \frac{1}{2} (\operatorname{tanh}(2) - \operatorname{tanh}(0))\]

Step 6: Evaluate the definite integral

Finally, evaluate the definite integral by calculating \(\operatorname{tanh}(2)\) and \(\operatorname{tanh}(0)\): \[ \frac{1}{2} (\operatorname{tanh}(2) - \operatorname{tanh}(0)) = \frac{1}{2} (\operatorname{tanh}(2) - 0)\] So the definite integral is: \[ \int_{0}^{4} \frac{\operatorname{sech}^{2} \sqrt{x}}{\sqrt{x}} \, dx = \frac{1}{2} \operatorname{tanh}(2) \]

Key Concepts

Substitution Method

The substitution method is a powerful technique used to simplify an integral by changing variables. This can make the integral easier to evaluate.
In this exercise, the substitution method involves letting a new variable, \( u \), replace an expression within the integrand. Here, we let \( u = \sqrt{x} \). This change of variables not only simplifies the integrand but also transforms the differential \( dx \) into a form that depends on \( du \).
Next, establish a relationship between \( x \) and \( u \): \( x = u^2 \), and differentiate it to find \( dx \). By differentiating, we find \( du = 2u \, dx \). Therefore, \( dx \) can be expressed as \( \frac{du}{2u} \).

• This substitution simplifies the computations.
• It can be crucial for integrating complex or unwieldy functions.

Bounds of Integration

When using substitution in a definite integral, it is vital to also transform the bounds of integration to match the new variable. This ensures that the integration process is coherent and consistent with the substitution made.
In our problem, the original bounds are from 0 to 4 for \( x \).

• The lower bound \( x = 0 \) becomes \( u = \sqrt{0} = 0 \).
• The upper bound \( x = 4 \) becomes \( u = \sqrt{4} = 2 \).

Thus, the transformation of bounds is a crucial step in maintaining the consistency of the substitution method. It ensures that the evaluation of the integral covers the intended interval in the transformed variable space.

Hyperbolic Functions

Hyperbolic functions, such as \( \operatorname{sech}(u) \) and \( \operatorname{tanh}(u) \), are analogues of trigonometric functions but for hyperbolas. In this exercise, the integrand involves \( \operatorname{sech}^2(u) \).
The integral \( \int \operatorname{sech}^2(u) \, du \) is known for having a direct antiderivative \( \operatorname{tanh}(u) \), which simplifies the integration step significantly.

• These functions have distinct properties beneficial in solving integrals involving hyperbolic terms.
• They play a similar role as trigonometric functions do for circles.

Understanding the derivatives and antiderivatives of hyperbolic functions is crucial for efficiently performing integration tasks like in this exercise.

Integral Simplification

Simplifying an integral involves transforming it into a form that's straightforward to evaluate. This often entails using various mathematical techniques to reduce complex expressions to simpler ones.
In this scenario, the original integral, after substitution and transformation, is \[ \int_{0}^{2} \frac{1}{2} \operatorname{sech}^2(u) \, du \]. The factor of \( \frac{1}{2} \) simplifies the integration process since it can be factored out, allowing focus on integrating \( \operatorname{sech}^2(u) \) alone.

• Factoring constants out of the integral can make calculations more manageable.
• Canceling out common terms reduces the complexity of the integrand.

Ultimately, these steps make the overall task of evaluating the integral much more efficient, as seen in the final integration and evaluation.