Problem 10 - Calculus Early Transcendentals: Pearson New International Edition

Question

Perform the indicated integrations. $$ \int_{0}^{4} \frac{5}{\sqrt{2 t+1}} d t $$

Step-by-Step Solution

Answer

The integral evaluates to 10.

Step 1: Identify the Type of Integration

The integral given is \( \int_{0}^{4} \frac{5}{\sqrt{2t+1}} \, dt \). This is an example of a definite integral with respect to \( t \), and will involve substitution because the integrand involves a composite function.

Step 2: Choose an Appropriate Substitution

Let \( u = 2t + 1 \). Then the derivative \( du = 2 \, dt \). To change \( dt \) to \( du \), we find \( dt = \frac{1}{2} du \). The limits of integration also change: when \( t = 0 \), \( u = 2(0) + 1 = 1 \); when \( t = 4 \), \( u = 2(4) + 1 = 9 \).

Step 3: Rewrite the Integral in Terms of the New Variable

Substitute \( u = 2t + 1 \) and \( dt = \frac{1}{2} du \) into the integral, changing the bounds as identified: \[ \int_{1}^{9} \frac{5}{\sqrt{u}} \, \frac{1}{2} du = \frac{5}{2} \int_{1}^{9} u^{-\frac{1}{2}} \, du. \]

Step 4: Integrate with Respect to \( u \)

Apply the power rule for integration: \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \). Here, \( n = -\frac{1}{2} \), so \( \int u^{-\frac{1}{2}} \, du = 2u^{\frac{1}{2}} \). Integrating, we get: \[ \frac{5}{2} \times 2u^{\frac{1}{2}} = 5u^{\frac{1}{2}}. \]

Step 5: Evaluate the Definite Integral

Evaluate \( 5u^{\frac{1}{2}} \) from \( u = 1 \) to \( u = 9 \): \[ 5 \left[ 9^{\frac{1}{2}} - 1^{\frac{1}{2}} \right] = 5 \left[ 3 - 1 \right] = 5 \times 2 = 10. \]

Key Concepts

Substitution Method

The substitution method is a key technique in calculus, particularly useful for evaluating integrals of composite functions. This involves changing the variable of integration to simplify the integrand.
In the given exercise, we start with the substitution \( u = 2t + 1 \). By choosing this substitution, we simplify the expression \( \frac{5}{\sqrt{2t+1}} \). The reason substitution is advantageous is because it helps convert a complex function into a simpler form.

• First, calculate the derivative: \( du = 2 \, dt \).
• Rearrange to express \( dt \) in terms of \( du \): \( dt = \frac{1}{2} \, du \).

When changing variables, it's also necessary to update the limits of integration. Initially given as \( t = 0 \) to \( t = 4 \), these need to change to \( u \) limits with the substitution:

• When \( t=0 \), \( u = 2(0) + 1 = 1 \).
• When \( t=4 \), \( u = 2(4) + 1 = 9 \).
With these new limits, the integral becomes easier to manage and can be solved in terms of \( u \) instead of \( t \).

Power Rule for Integration

The power rule for integration is one of the fundamental techniques used to find antiderivatives of power functions. This rule states that for any real number \( n eq -1 \), \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \).
Applying this rule in the context of our problem, we integrate the function \( u^{-\frac{1}{2}} \), intending to simplify it for evaluation. The exponent here is \( n = -\frac{1}{2} \). According to the power rule:

• \( \int u^{-\frac{1}{2}} \, du = \frac{u^{\frac{1}{2}}}{\frac{1}{2}} \).
• This simplifies to \( 2u^{\frac{1}{2}} \).

Now, our integral becomes \( \frac{5}{2} \times 2u^{\frac{1}{2}} = 5u^{\frac{1}{2}} \). This computes the antiderivative of the simplified integrand, making it ready for the final step of evaluation.

Changing Limits of Integration

Changing the limits of integration is essential when you substitute variables in a definite integral. This step ensures that the resulting integral remains accurate with the new variable.
Initially, the integral for \( t \) from 0 to 4 was transformed into an integral for \( u \).
As derived:

• When \( t = 0 \), \( u = 1 \).
• When \( t = 4 \), \( u = 9 \).

These new limits, \( u = 1 \) to \( u = 9 \), become crucial since they dictate the portion of the function that the integral computes over. After the power rule was applied, we are left with: \( 5u^{\frac{1}{2}} \). The final step is evaluating this expression using these changed limits:

• Calculate \( 5 \times [ 9^{\frac{1}{2}} - 1^{\frac{1}{2}} ] \).
• Since \( 9^{\frac{1}{2}} = 3 \) and \( 1^{\frac{1}{2}} = 1 \), we have: \( 5(3-1) = 10 \).

Thus, the correct computation hinges on using the adapted limits from the substitution method, concluding the evaluation of the integral.