Problem 106 - Calculus and its applications

Question

Evaluate. $$ \int \frac{t^{2}+2 t}{(t+1)^{2}} d t(\text { Hint: } u=t+1 .) $$

Step-by-Step Solution

Answer

The integral evaluates to \( t + 1 + \frac{1}{t+1} + C \).

Step 1: Set up the substitution

We are given a hint: use the substitution \( u = t + 1 \). This means \( du = dt \) and \( t = u - 1 \). The integral becomes: \[ \int \frac{(u-1)^2 + 2(u-1)}{u^2} \, du. \] Substituting these expressions into the integral, we can perform the integration in terms of \( u \).

Step 2: Simplify the integrand

Expand the expression in the numerator: \((u-1)^2 + 2(u-1) = u^2 - 2u + 1 + 2u - 2 = u^2 - 1\). So the integral becomes: \[ \int \frac{u^2 - 1}{u^2} \, du. \] Simplifying the fraction, we obtain: \[ \int \left(1 - \frac{1}{u^2}\right) \, du. \]

Step 3: Integrate term by term

Now we integrate each term separately: \[ \int 1 \, du = u \] and \[ \int -\frac{1}{u^2} \, du = \int -u^{-2} \, du = -(-u^{-1}) = \frac{1}{u}. \] So, the integral simplifies to \( u + \frac{1}{u} + C \), where \( C \) is the constant of integration.

Step 4: Substitute back to original variable

Replace \( u \) with the original variable expression \( t+1 \): \[ (t + 1) + \frac{1}{t + 1} + C. \] This is the evaluation of the integral in terms of \( t \).

Key Concepts

Integral Calculus

Integral Calculus is a branch of calculus focusing on the process of integration, which is essentially the reverse operation of differentiation. It deals with finding the whole given its parts, and can be used to determine areas, volumes, and even solutions to differential equations. In simple terms:

• Integration can be seen as summing up infinitely small slices to find the total.
• It's a key concept in mathematics, particularly for defining functions not easily expressible using elementary algebra.

In this exercise, we deal with a common problem in integral calculus, which is to evaluate a definite or indefinite integral. Here, we aim to simplify the given problem using the substitution method, an effective integration technique.

Integration Techniques

There are various techniques for solving integrals, and the choice depends on the form of the integrand.

• Basic Techniques: These include finding integrals with basic antiderivatives such as power rule, product rule, and sum rule.
• Advanced Techniques: These involve integration by parts, partial fractions decomposition, and trigonometric substitution.
• The choice of technique depends on the complexity and structure of the function being integrated.

For the current problem, the substitution method is optimal. Substitution transforms the integral into a simpler form that is easier to evaluate.

Substitution Method

The substitution method, also known as u-substitution, is a technique used in integration to simplify the integration process. This method is akin to the chain rule for derivatives, where a substitution is made to transform a complex function into a simpler one.

• Identify a part of the integrand that can be set as a new variable, typically denoted as "u".
• Express all parts of the integrand in terms of "u", including the differential "dt" changing to "du".
• Integrate the simpler "u" function. This often involves straightforward antiderivatives.
• Finally, substitute back in terms of the original variable to complete the process.

In this exercise, the substitution of "u = t + 1" was selected. This change simplifies the integrand and streamlines the integration.