Problem 59 - Thomas Calculus

Question

Evaluate the integrals. \(\int_{0}^{3}(\sqrt{2}+1) x^{\sqrt{2}} d x\)

Step-by-Step Solution

Answer

The value of the integral is \( 3^{\sqrt{2}+1} \).

Step 1: Identify the integral form

The given integral is \( \int_{0}^{3} (\sqrt{2}+1) x^{\sqrt{2}} \, dx \). We recognize that this is an integral of the form \( \int x^n \, dx \), where \( n = \sqrt{2} \), and \( \sqrt{2} + 1 \) is a constant factor.

Step 2: Apply the power rule for integration

The power rule for integration states that \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration. For our problem, since we are evaluating a definite integral, the constant term is not needed. The integral transforms to \( (\sqrt{2} + 1) \left[ \frac{x^{\sqrt{2} + 1}}{\sqrt{2} + 1} \right] \).

Step 3: Simplify the integral expression

Since \( \sqrt{2} + 1 \) is a constant, it can be factored outside the integral. This gives \[ (\sqrt{2}+1) \left[ \frac{x^{\sqrt{2}+1}}{\sqrt{2}+1} \right]_{0}^{3} = \left[ x^{\sqrt{2}+1} \right]_{0}^{3} \].

Step 4: Evaluate the definite integral

Now, evaluate the expression \( \left[ x^{\sqrt{2}+1} \right]_{0}^{3} \). Plug in the limits of integration, we get: \[ \left[ 3^{\sqrt{2}+1} - 0^{\sqrt{2}+1} \right] = 3^{\sqrt{2}+1} \] since \( 0^{\sqrt{2}+1} = 0 \).

Step 5: Finalize the calculation

The result of the integral is: \( 3^{\sqrt{2}+1} \). This is the final value of the definite integral from 0 to 3.

Key Concepts

Power Rule for Integration

The power rule for integration is a fundamental concept in calculus used to find the antiderivative of a function when dealing with polynomials. It says that when you have an integral in the form \( \int x^n \, dx \), you can find its integral by using the formula \( \frac{x^{n+1}}{n+1} + C \), where \( n \) is not equal to \(-1\) and \( C \) is the constant of integration. This rule helps us to find the area under a curve, especially in simple polynomial cases.
In the given exercise, the function \( x^{\sqrt{2}} \) is integrated using this power rule. Here, \( n \) is equal to \( \sqrt{2} \), so applying the power rule changes the expression to \( \frac{x^{\sqrt{2} + 1}}{\sqrt{2} + 1} \). Remember, with definite integrals, you usually do not add the constant \( C \) because you're computing a specific value between two limits.

Definite Integral Evaluation

Evaluating a definite integral involves calculating the net area under the curve of a function between two specified limits. This process gives a specific numerical outcome, unlike indefinite integrals, which provide a general antiderivative function. In the definite integral \( \int_{a}^{b} f(x) \, dx \), the lower limit \( a \) and upper limit \( b \) are where you start and end measuring the accumulation of area.
In our exercise, after applying the power rule, the expression \( \left[ \frac{x^{\sqrt{2} + 1}}{\sqrt{2} + 1} \right]_{0}^{3} \) is evaluated by substituting the upper limit 3 and lower limit 0. The outcome calculates the net area from 0 to 3 for the function \( x^{\sqrt{2}} \). Performing the subtraction \( f(3) - f(0) \) results in \( 3^{\sqrt{2}+1} - 0 \), simplifying to \( 3^{\sqrt{2}+1} \). This final value represents the total accumulated area from the start to the end of the interval.

Exponential Function Integration

Integrating exponential functions often requires different approaches based on the form of the exponential expression. However, in the context of this exercise, the function \( x^{\sqrt{2}} \) is addressed similarly to polynomial expressions, treated through the power rule. While \( x^{\sqrt{2}} \) is not an exponential in the classical sense like \( e^x \), the integration process here is straightforward due to the rational nature of \( \sqrt{2} \).
To evaluate, the first step was to use the power rule, which aligns with how bases of variable exponentials behave. Once you apply the rule, what's left is accurate evaluation of the definite integral to determine its exact value. Understanding this helps grasp how to transition from abstract variable expressions to finite numerical results, reinforcing the ability to systematically handle expressions involving irrational exponents.