Problem 31 - Calculus: Early Transcendental Functions

Question

Evaluate the definite integral. $$\int_{0}^{2} x \sqrt{x^{2}+1} d x$$

Step-by-Step Solution

Answer

The value of the definite integral \( \int_{0}^{2} x \sqrt{x^{2}+1} dx \) is \( \frac{1}{3} \left[5 \sqrt{5}-1\right] \).

Step 1: Recognize the Pattern

Look at the integral \( \int_{0}^{2} x \sqrt{x^{2}+1} dx \). This is a case for using substitution because \( \sqrt{x^{2}+1} \) is the derivative of \( x \). So set \( u=x^{2}+1 \). Then the derivative \( du = 2x dx \). Therefore, we can write \( x dx = \frac{1}{2} du \). Before we substitute, it is necessary to change the limits of integration from x to u. When \( x=0, u=0^{2}+1=1 \) and when \( x=2, u=2^{2}+1=5 \).

Step 2: Perform Substitution

Substitute u values, du values and the new boundaries back to the integral, transforming \( \int_{0}^{2} x \sqrt{x^{2}+1} dx \) into \( \int_{1}^{5} \sqrt{u} \left(\frac{1}{2}\right) du \). Simplify it as \( \frac{1}{2} \int_{1}^{5} u^{\frac{1}{2}} du \).

Step 3: Calculate the Integral

Calculate the integral \( \frac{1}{2} \int_{1}^{5} u^{\frac{1}{2}} du \). The antiderivative of \( u^{\frac{1}{2}} \) is \( \frac{2}{3}u^{\frac{3}{2}} \). Therefore, \( \frac{1}{2} \int_{1}^{5} u^{\frac{1}{2}} du = \frac{1}{2} \cdot \frac{2}{3} \left[u^{\frac{3}{2}}\right]_{1}^{5} = \frac{1}{3} \left[\left(5^{\frac{3}{2}}\right)-\left(1^{\frac{3}{2}}\right)\right] \).

Step 4: Compute the Value

Finally do the computation: \( \frac{1}{3} \left[\left(5^{\frac{3}{2}}\right)-\left(1^{\frac{3}{2}}\right)\right] = \frac{1}{3} \left[5 \sqrt{5}-1\right] \).

Key Concepts

Substitution Method

The substitution method is a powerful technique for evaluating integrals, especially when dealing with functions that seem complex at first glance. The idea is to simplify the integral by introducing a new variable. In this exercise, we look at the integral \( \int_{0}^{2} x \sqrt{x^{2}+1} \, dx \). Here, the expression \( \sqrt{x^{2}+1} \) allows us to use substitution, as it naturally relates to a simpler function's derivative.

The substitution works by setting \( u = x^{2}+1 \). The next step is finding the differential \( du \). As \( u \) depends on \( x \), its derivative becomes \( du = 2x \, dx \).
This helps us to rewrite the original integral in terms of \( u \) and \( du \). Thus, \( x \, dx \) is expressed as \( \frac{1}{2} \, du \).
The substitution transforms the problem into something simpler and often more manageable. By making the substitution, the integral becomes \( \frac{1}{2} \int \sqrt{u} \, du \), now free of the initial variable \( x \), allowing for easier integration.

Integration Limits

Whenever substitution is used in a definite integral, it is crucial to also adjust the integration limits to reflect the new variable. It ensures you are evaluating the integral over the correct interval. In this example, converted limits are necessary due to the substitution of \( u = x^{2} + 1 \).

For the original integral \( \int_{0}^{2} \, x \sqrt{x^{2}+1} \, dx \), the variable \( x \) varies between 0 and 2.
The new limits for \( u \) must be calculated as follows:

• When \( x = 0 \), \( u = 0^2 + 1 = 1 \).
• When \( x = 2 \), \( u = 2^2 + 1 = 5 \).

Thus, the integral in terms of \( u \) becomes \( \int_{1}^{5} \sqrt{u} \, du \).
Adjusting limits is an important step in using substitution which guarantees the evaluation remains valid within the new context.

Antiderivative Calculation

The calculation of an antiderivative is a fundamental component of solving integrals. After substitution, we need to find the antiderivative of the new expression. For our specific integral \( \frac{1}{2} \int_{1}^{5} u^{\frac{1}{2}} \, du \), the focus is on the function \( u^{\frac{1}{2}} \).

The antiderivative, also known as the indefinite integral, is determined through the reverse process of differentiation. When integrating power functions, we use the formula:\[ \int u^{n} \, du = \frac{u^{n+1}}{n+1} \] Applying this to \( u^{\frac{1}{2}} \):

• Add one to the exponent: \( \frac{1}{2} + 1 = \frac{3}{2} \).
• Divide by the new exponent: \( \frac{u^{\frac{3}{2}}}{\frac{3}{2}} = \frac{2}{3}u^{\frac{3}{2}} \).

This makes our antiderivative \( \frac{2}{3}u^{\frac{3}{2}} \).
The antiderivative calculation is the stepping stone needed to solve the definite integral.

Integral Evaluation Steps

Once the antiderivative is determined, evaluating the definite integral involves a sequence of straightforward steps. It transforms the integral into a calculable numeric value by substituting the integration limits.

Working with \( \frac{1}{2} \int_{1}^{5} u^{\frac{1}{2}} \, du \), the process includes:

• Using the antiderivative \( \frac{2}{3}u^{\frac{3}{2}} \).
• Multiply by \( \frac{1}{2} \), as factored out earlier. This gives \( \frac{1}{3}u^{\frac{3}{2}} \).
• Substitute the upper limit (5) and lower limit (1) into \( \frac{1}{3}u^{\frac{3}{2}} \).
• Compute: \( \frac{1}{3} \left[5^{\frac{3}{2}} - 1^{\frac{3}{2}}\right] \).
• This simplifies to \( \frac{1}{3} \left[5\sqrt{5} - 1\right] \).

By following these steps, the integral simplifies to a specific number, thus concluding the evaluation process.