Problem 16 - Applied Calculus

Question

Find the integrals in problems. Check your answers by differentiation. $$ \int(2 t-7)^{73} d t $$

Step-by-Step Solution

Answer

The integral is \( \frac{(2t-7)^{74}}{148} + C \).

Step 1: Identify the Type of Integral

The integral we have is \( \int(2t-7)^{73} \, dt \). This integral looks like a power of a linear function, which suggests we can use the power rule for integration along with a substitution method.

Step 2: Choose a Substitution

Let \( u = 2t - 7 \). Then, differentiate \( u \) with respect to \( t \) to find \( \frac{du}{dt} = 2 \). Rearranging gives us \( dt = \frac{1}{2} du \).

Step 3: Substitute in the Integral

Now, replace \( (2t-7) \) with \( u \) and \( dt \) with \( \frac{1}{2} du \) in the original integral: \[\int (2t-7)^{73} \, dt = \int u^{73} \frac{1}{2} du \]This simplifies to:\[ \frac{1}{2} \int u^{73} \, du \]

Step 4: Apply the Power Rule

To integrate \( \frac{1}{2} \int u^{73} \, du \), use the power rule \( \int u^n du = \frac{u^{n+1}}{n+1} + C \). Applying it here gives:\[ \frac{1}{2} \cdot \frac{u^{74}}{74} + C = \frac{u^{74}}{148} + C \]

Step 5: Back-Substitute \( u \)

Replace \( u \) back with the original expression \( 2t - 7 \) to express the antiderivative in terms of \( t \):\[ \frac{(2t-7)^{74}}{148} + C \]

Step 6: Differentiate to Check the Answer

Differentiate the result \( \frac{(2t-7)^{74}}{148} + C \) and confirm it matches the original integrand:Using the chain rule, the derivative is:\[ \frac{74(2t-7)^{73}}{148} \cdot 2 = (2t-7)^{73} \]This confirms the integration was performed correctly since the derivative matches the original integrand.

Key Concepts

Substitution Method

The substitution method is an essential technique in calculus integration. It simplifies integrals by changing variables, transforming a complex integral into a more manageable form. Let's consider the integral \( \int(2t-7)^{73} \, dt \). Here, the substitution method is ideal because of the composition of a linear function and a power.

• First, choose a substitution. We let \( u = 2t - 7 \) because it simplifies the expression. This transforms the integral's variable from \( t \) to \( u \), making the power rule easier to apply.
• Next, find the relationship between \( du \) and \( dt \). Differentiate \( u \) with respect to \( t \) to find \( \frac{du}{dt} = 2 \). Solving for \( dt \), we get \( dt = \frac{1}{2} du \).
• Replace \( (2t-7) \) with \( u \) and \( dt \) with \( \frac{1}{2} du \) in the integral. This substitution changes the original integral to \( \frac{1}{2} \int u^{73} \, du \).

By using the substitution method, we are able to convert the complex integrand into a form where the power rule can be effectively applied, simplifying the process of integration.

Power Rule Integration

Power rule integration is a straightforward technique used to integrate functions of the form \( u^n \). Once you've used the substitution method to transform your integral, applying the power rule becomes simple. Our transformed integral is \( \frac{1}{2} \int u^{73} \, du \).

• The power rule for integration states: \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.
• Apply the power rule to \( u^{73} \): Integrate it to get \( \frac{u^{74}}{74} \).
• Incorporate the factor \( \frac{1}{2} \) from earlier substitution: Multiply to obtain \( \frac{u^{74}}{148} + C \).

The power rule simplifies the integration process for polynomial expressions. By applying it correctly, the definite integral is found smoothly and accurately.

Differentiation Check

Once the integration is complete, it's essential to verify the correctness of your result. This verification is done through differentiation. After integrating and substituting back, the result was \( \frac{(2t-7)^{74}}{148} + C \).

• Differentiate the result with respect to \( t \) to check against the original integrand. Use the chain rule to differentiate \( \frac{(2t-7)^{74}}{148} + C \).
• The derivative is \( \frac{74(2t-7)^{73}}{148} \cdot 2 \). Simplify it to reach \( (2t-7)^{73} \).

Matching the derived expression to the original integrand \( (2t-7)^{73} \) confirms the accuracy of the integration. Checking through differentiation ensures that no errors occurred during integration, providing confidence in the solution.