Problem 37 - Calculus

Question

Use the Substitution Rule for Definite Integrals to evaluate each definite integral. $$ \int_{-1}^{3} \frac{1}{(t+2)^{2}} d t $$

Step-by-Step Solution

Answer

The value of the integral is \( \frac{4}{5} \).

Step 1: Identify the substitution

We need to choose a substitution that simplifies the integrand. Let's use the substitution \( u = t + 2 \), which will simplify the denominator.

Step 2: Find the derivative and change of variables

Differentiate the substitution: \( du = dt \). This means our integral will be in terms of \( u \), and \( du \) replaces \( dt \).

Step 3: Change the limits of integration

With the substitution \( u = t + 2 \), calculate the new limits: when \( t = -1 \), \( u = -1 + 2 = 1 \); when \( t = 3 \), \( u = 3 + 2 = 5 \). Thus, the new limits for \( u \) are from 1 to 5.

Step 4: Substitute and solve the integral

Substitute into the integral: \( \int_{1}^{5} \frac{1}{u^2} \, du \). This integral simplifies to \( \int_{1}^{5} u^{-2} \, du \). The integral of \( u^{-2} \) is \( -u^{-1} \), or \( -\frac{1}{u} \).

Step 5: Evaluate the definite integral

Evaluate \( -\frac{1}{u} \) from 1 to 5: \( \left[-\frac{1}{u}\right]_1^5 = \left(-\frac{1}{5}\right) - \left(-\frac{1}{1}\right) = -\frac{1}{5} + 1 = \frac{4}{5} \).

Key Concepts

Definite Integral

A definite integral represents the area under a curve, usually between two specific points on the x-axis. In simpler terms, it's a number that gives us a clear value of this area.
In our problem, we are given the definite integral from \(-1\) to \(3\). That means we want to find the area under the curve \(\frac{1}{(t+2)^2}\), from \(t = -1\) to \(t = 3\).
Here are key takeaways:

• Definite integrals have limits or boundaries; these are the numbers at the bottom and top of the integral sign.
• The result is a specific numerical value, rather than a function.

Understanding the definite integral helps us find the total accumulated value between given limits.

Integration by Substitution

Integration by substitution is like the reverse of the chain rule in differentiation. It allows us to simplify complicated integrands by making a substitution.
In the given exercise, our integral \(\int_{-1}^{3} \frac{1}{(t+2)^2} \, dt\) becomes simpler with the substitution \(u = t + 2\).
When doing substitution, keep in mind:

• Choose a substitution \(u\) that simplifies your integral, usually something inside a parenthesis or an exponent.
• Differentiate \(u\) with respect to \(t\) to get \(du\). This helps in expressing everything in terms of \(u\).
• Change the limits of integration according to your substitution. This ensures your definite integral's boundaries are accurate and consistent with the new variable.

Substitution transforms a challenging problem into an easier form, allowing us to integrate more smoothly.

Calculus Problem Solving

Solving calculus problems can feel daunting, but breaking them down into clear, manageable steps is key.
For the exercise at hand, we used substitution to simplify the integral, and then transformed it back into a solvable form.
Here's a simple strategy:

• Identify and simplify: Look for patterns or substitutions that simplify your integrals.
• Do the math: Follow through with the arithmetic; carry out the integration then apply limits precisely.
• Double-check: Once solved, reassess your working and calculations to ensure everything is accurate.

Effective problem solving in calculus involves accuracy, logical sequences, and sometimes a bit of creative thinking to simplify and solve the problems.