Problem 48 - Calculus and its applications

Question

Evaluate. $$ \int \frac{t e^{t}}{(t+1)^{2}} d t $$

Step-by-Step Solution

Answer

The integral evaluates to \( 2 \frac{e^{t}}{t+1} + C \).

Step 1: Choose the Appropriate Technique

To tackle the integral \( \int \frac{t e^{t}}{(t+1)^{2}} \ dt \), notice that it involves a rational function of a linear term and an exponential function. This suggests using integration by parts or substitution. We elect to proceed with substitution as a simplification mechanism.

Step 2: Define a Substitution

Choose the substitution \( u = t + 1 \). This gives us \( du = dt \) and \( t = u - 1 \). Substituting back into the integral gives:\[ \int \frac{(u-1)e^{u-1}}{u^2} \, du \]

Step 3: Simplify the Integral

Distribute and simplify:\[ \int \left( \frac{ue^{u-1}}{u^2} - \frac{e^{u-1}}{u^2} \right) \, du = \int \left( \frac{e^{u-1}}{u} - \frac{e^{u-1}}{u^2} \right) \, du \]

Step 4: Recognize Integration by Parts

To integrate \( \int \frac{e^{u-1}}{u} \, du \), we use integration by parts. Let \( v = \frac{1}{u} \), \( dv = -\frac{1}{u^2} \, du \), \( dw = e^{u-1} \, du = e^{u}e^{-1} \, du \), \( w = e^{u} \).

Step 5: Apply Integration by Parts

Using integration by parts formula \( \int v \, dw = vw - \int w \, dv \):\[ \int \frac{e^{u}}{ue} \, du = \left[ \frac{e^{u}}{u} \right] - \int \left( \frac{-e^{u}}{u^2} \right) \, du \]

Step 6: Evaluate the Second Integral

The second integral \( \int \frac{-e^{u}}{u^2} \) is already in the integrable form after simplification from the original:\[ \int \frac{e^{u-1}}{u^2} \, du = \int \frac{e^{u} \, e^{-1}}{u^2} \, du = -\frac{e^{u}}{u} \]

Step 7: Combine Results

Combine the results:\[ \frac{e^{u}}{u} - \int \frac{e^{u}}{u^2} \, du = \frac{e^{u}}{u} + \frac{e^{u}}{u} = \frac{2e^{u}}{u} \]

Step 8: Reverse Substitution

Revert \( u = t + 1 \):\[ \frac{2e^{t+1}}{t+1} = 2 \frac{e^{t}}{t+1} \]

Step 9: Evaluate the Constant of Integration

With any indefinite integral, add the constant of integration \( + C \). Thus, the final answer is:\[ 2 \frac{e^{t}}{t+1} + C \]

Step 10: Final Answer

The evaluated integral \( \int \frac{t e^{t}}{(t+1)^2} \, dt \) is:\[ 2 \frac{e^{t}}{t+1} + C \] where \( C \) is the constant of integration.

Key Concepts

Substitution Method

The substitution method, also known as "u-substitution," is a common technique in integral calculus used to simplify complex integrals. The principle behind this method involves transforming the variable of integration to simplify the integrand into a form that is easier to handle.

To apply the substitution method:

• Select a substitution that will simplify the integral. Typically, this involves choosing a part of the integrand that can be redefined using a new variable.
• For this problem, we chose the substitution \( u = t + 1 \). This makes the derivative \( du = dt \), and consequently, we have \( t = u - 1 \).
• Substitute all occurrences of the original variable in the integrand, replacing it with \( u \) and its corresponding differential. In this case, the integral \( \int \frac{t e^{t}}{(t+1)^{2}} \, dt \) transforms into \( \int \frac{(u-1)e^{u-1}}{u^2} \, du \).

This substitution effectively simplifies the rational function of the integrand, making it more straightforward to apply further integration techniques like integration by parts.

Integration by Parts

Integration by parts is another fundamental method used in calculus to integrate products of functions. It is derived from the product rule of differentiation and helps when dealing with integrands that are products of functions.

The integration by parts formula is \( \int u \, dv = uv - \int v \, du \). To use this technique effectively:

• Choose which parts of the original integral will represent \( u \) and \( dv \). Application usually depends on simplifying the integrand in subsequent steps.
• In this problem, after substituting, we found ourselves needing to integrate \( \int \frac{e^{u}}{ue} \, du \). Here, we set \( v = \frac{1}{u} \) and \( dv = -\frac{1}{u^2} \, du \), while \( dw = e^{u-1} \, du = e^{u}e^{-1} \, du \), and so, \( w = e^{u} \).
• Apply the formula to transform the integral into more manageable parts: \( \left[ \frac{e^{u}}{u} \right] - \int \left( \frac{-e^{u}}{u^2} \right) \, du \).

After applying integration by parts, the integral simplifies further, making it easier to evaluate and combine with other parts of the problem.

Constant of Integration

The constant of integration plays a crucial role in indefinite integrals. When an antiderivative of a function is found, it represents a family of functions that differ by a constant. Since the derivative of any constant is zero, integrating always involves adding an arbitrary constant.

• In the context of our problem, after performing integration and combining all parts, we arrive at a mathematical expression representing the general form of antiderivatives for the integrand.
• The solution is \( 2 \frac{e^{t}}{t+1} \), but it only reflects one member of a family of solutions. To account for all possible antiderivatives, we add the constant \( C \), yielding the final result: \( 2 \frac{e^{t}}{t+1} + C \).

This constant is critical because it embodies the infinite possibilities that arise from solving indefinite integrals, allowing for accurate representation of the function’s behavior across different scenarios or initial conditions.