Problem 92 - Thomas Calculus

Question

Evaluate the integrals. \begin{equation}\int_{1}^{4} \frac{2^{\sqrt{x}}}{\sqrt{x}} d x\end{equation}

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{4}{\ln(2)} \).

Step 1: Identify the Type of Integral

The integral is \[ \int_{1}^{4} \frac{2^{\sqrt{x}}}{\sqrt{x}} \, dx \]This appears to be an integral of the form involving a substitution to simplify the exponent and the denominator.

Step 2: Choose an Appropriate Substitution

Let \( u = \sqrt{x} \), thereby \( x = u^2 \). Calculate the derivative of this substitution: \( dx = 2u \, du \). The limits of integration change from \( x=1 \) to \( u=1 \) and \( x=4 \) to \( u=2 \).

Step 3: Perform the Substitution

Substitute \( u = \sqrt{x} \) and \( dx = 2u \, du \) into the integral. Change the limits accordingly:\[ \int_{1}^{2} \frac{2^u}{u} \cdot 2u \, du = \int_{1}^{2} 2^{u+1} \, du \]

Step 4: Integrate the Simplified Expression

The integral becomes\[ \int 2^{u+1} \, du = \int 2 \cdot 2^u \, du = 2 \cdot \frac{2^u}{\ln(2)} + C \]based on the formula for integrating an exponential function \( \int a^x \, dx = \frac{a^x}{\ln(a)} + C \).

Step 5: Evaluate the Definite Integral

Substitute the limits of integration back into the result:\[ 2 \cdot \left( \frac{2^u}{\ln(2)} \right) \bigg|_1^2 = \frac{2 \cdot 2^2}{\ln(2)} - \frac{2 \cdot 2^1}{\ln(2)} = \frac{8}{\ln(2)} - \frac{4}{\ln(2)} = \frac{4}{\ln(2)} \]

Step 6: Present the Final Answer

The value of the integral \( \int_{1}^{4} \frac{2^{\sqrt{x}}}{\sqrt{x}} \, dx \) is \( \frac{4}{\ln(2)} \).

Key Concepts

Substitution Method

The substitution method is a powerful technique for solving integrals that are not easily approachable in their original form. It involves changing variables in the integral to simplify the expression. This method is particularly useful for integrals involving compositions, where one function is nested inside another, or when terms seem complex related to logarithms or exponentials.

To apply the substitution method, follow these steps:

• Identify a substitution that can simplify the integral. This often involves setting a variable, like \( u \), to a part of the integrand that, when differentiated, corresponds to another part of the integrand.
• Calculate the differential of your substitution, \( dx \), which will be used to replace \( dx \) in the integral.
• Change the limits of integration if you're working with definite integrals to align with your new variable.
• Substitute into the integral with all expressions replaced by your new variable and its differential.
• Integrate with respect to the new variable and, if necessary, revert back to the original variable for the final answer.

By simplifying integrations through substitutions, you essentially yield a form that might match standard integral formulas already known.

Exponential Integrals

Exponential integrals involve expressions where the variable of integration appears as an exponent. They often present unique challenges because of how exponents interact with differentiation and integration. For these integrals, substitution can transform an otherwise difficult problem into a manageable one.

The key to integrating exponentials is often recognizing it in the form \( a^x \), where the base is a constant. The integral of \( a^x \), where \( a \) is a constant and not equal to 1, can be expressed as:\[ \int a^x \, dx = \frac{a^x}{\ln(a)} + C \]
When integrating typical exponential functions, the natural logarithm \( \ln(a) \) appears because it is the correction factor stemming from the derivative of an exponential function relative to a non-natural base.

In the substitution method when the integrand is an exponential term like \( 2^u \), you seek a transformation that will isolate terms making it easier to perform the integration, as illustrated in the step-by-step solution. Understanding these properties of exponential integrals is beneficial for proficiency in calculus and tackling complex calculus problems.

Definite Integrals

Definite integrals provide the area under a curve from one point to another, which is a central concept in calculus and has applications in physics, economics, and engineering. They differ from indefinite integrals, which represent a family of functions, as definite integrals compute a specific numerical value.

To solve a definite integral, follow these guidelines:

• Determine the antiderivative of the function to apply the Fundamental Theorem of Calculus.
• Evaluate the antiderivative at the upper limit and lower limit of integration, denoted as \( F(b) - F(a) \), where \( b \) and \( a \) are the upper and lower bounds, respectively.
• Subtract the value obtained by substituting the lower limit from the value obtained by substituting the upper limit.

This process calculates the total accumulation or net change represented by the integral.

When performing substitution along with definite integrals, don't forget that the limits of integration will change corresponding to the new variable. Ensuring you adjust these limits correctly is crucial to obtaining the correct area under the curve or solved value. These techniques highlight the analytical beauty of integral calculus and its many real-world applications.