Problem 21 - Applied Calculus

Question

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

Step-by-Step Solution

Answer

The integral is \(\frac{1}{148} (2t - 7)^{74} + C\).

Step 1: Recognize the Integral Form

The integral involved is of the form \( \int (2t - 7)^{73} \, dt \). This can be solved using a substitution method because it involves a power of a linear expression.

Step 2: Choose Substitution

Let \( u = 2t - 7 \). Then, the derivative is \( \frac{du}{dt} = 2 \), so \( dt = \frac{1}{2} du \). Substitute \( t \) terms with \( u \) terms.

Step 3: Rewrite the Integral

Substituting \( u = 2t - 7 \) and \( dt = \frac{1}{2} du \) into the original integral gives:\[\int (2t - 7)^{73} \, dt = \int u^{73} \cdot \frac{1}{2} \, du\]

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

The integral becomes:\[\frac{1}{2} \int u^{73} \, du\]Use the power rule for integration: \( \int u^n \, du = \frac{u^{n+1}}{n+1} + C \). Apply this:\[\frac{1}{2} \cdot \frac{u^{74}}{74} + C\]

Step 5: Simplify the Expression

Multiply the constants:\[\frac{1}{2} \cdot \frac{1}{74} = \frac{1}{148}\]The integral becomes:\[\frac{1}{148} u^{74} + C\]

Step 6: Back-Substitute \( u \)

Replace \( u \) with \( 2t - 7 \) to return to the variable \( t \):\[\int (2t - 7)^{73} \, dt = \frac{1}{148} (2t - 7)^{74} + C\]

Step 7: Verify by Differentiation

Differentiate \( \frac{1}{148} (2t - 7)^{74} + C \) with respect to \( t \) to check the result:\[\frac{d}{dt}\left(\frac{1}{148} (2t - 7)^{74}\right) = \frac{1}{148} \cdot 74 \cdot (2t - 7)^{73} \cdot 2 = (2t - 7)^{73}\]The derivative matches the integrand, confirming the solution.

Key Concepts

Substitution Method

The substitution method is a key technique in integration, often used when dealing with integrals that involve complex expressions. It simplifies the process by introducing a new variable. In our example, the integral \[ \int (2t - 7)^{73} \, dt \]is ideal for substitution because it features a power of a linear expression.
By letting \( u = 2t - 7 \), we can find \( du \) by differentiating \( u \) with respect to \( t \), which gives \( \frac{du}{dt} = 2 \). This leads to \( dt = \frac{1}{2} du \).
We substitute

• \( u \) for \( 2t - 7 \)
• \( \frac{1}{2} du \) for \( dt \)

This substitution transforms the integral into a simpler form: \[ \frac{1}{2} \int u^{73} \, du \].
This simplification makes it easier to apply the next technique, the power rule for integration.

Power Rule for Integration

The power rule is a fundamental tool in calculus, particularly useful when integrating expressions of the form \( u^n \). It states that:\[ \int u^n \, du = \frac{u^{n+1}}{n+1} + C \]where \( n eq -1 \) and \( C \) represents the constant of integration.
In our exercise, after applying the substitution method, our task is to integrate:\[ \frac{1}{2} \int u^{73} \, du \].
Using the power rule, this becomes \[ \frac{1}{2} \cdot \frac{u^{74}}{74} + C \].
This is then simplified by multiplying the constants together:\[ \frac{1}{148} u^{74} + C \].
Finally, substitute back \( u = 2t - 7 \) to express the solution in terms of the original variable \( t \):\[ \frac{1}{148} (2t - 7)^{74} + C \].
This process wraps up the integration part of solving the problem.

Differentiation for Verification

Verifying your integration result is an essential step to ensure its accuracy. The best way to do this is by differentiation. For our solved integral, \[ \frac{1}{148} (2t - 7)^{74} + C \],we differentiate it with respect to \( t \).
Start by applying the chain rule to differentiate \( (2t - 7)^{74} \). The chain rule tells us to:

• Differentiate the outer function: \( 74(2t - 7)^{73} \)
• Multiply by the derivative of the inner function, \( 2 \)

Thus, the derivative of \( \frac{1}{148} (2t - 7)^{74} \) is:\[ \frac{1}{148} \cdot 74 \cdot (2t - 7)^{73} \cdot 2 \].
On simplifying, this expression becomes \( (2t - 7)^{73} \), which perfectly matches the original integrand.
This confirms the correctness of our integration step. Such verification builds confidence in the method applied and the soundness of the result.