Problem 31 - College Algebra and Calculus: An Applied Approach

Question

Use a symbolic integration utility to find the indefinite integral. $$ \int\left(1+\frac{4}{t^{2}}\right)^{2}\left(\frac{1}{t^{3}}\right) d t $$

Step-by-Step Solution

Answer

The indefinite integral \( \int \frac{1}{\sqrt{x}} e^{\sqrt{x}} d x \) evaluates to \( 2e^{\sqrt{x}} \) + C.

Step 1: Variable Substitution

Identify the inner function within the integral that complicates the computation. In this case, the inner function can be identified as \( u= \sqrt{x} \). Hence, the integral changes its form. Also, calculate the derivative of 'u' to substitute for \( dx \), this gives \( du=\frac{1}{2\sqrt{x}}dx \) or \( dx= 2udu \). Now substitute 'u' and 'du' into the original integral to simplify it.

Step 2: Simplify the Integral Expression

After the substitution, the integral becomes \( 2\int e^{u} du \). This one is much easier to handle because the integral of an exponential function is itself.

Step 3: Evaluate the Integral

Evaluate the simplified integral to get \( 2e^{u} \) + C, where C is the constant of integration.

Step 4: Reverse Substitution

Finally, substitute back \( u= \sqrt{x} \) into the integral to get the answer in terms of x: \( 2e^{\sqrt{x}} \) + C.

Key Concepts

The Magic of Variable Substitution

Variable substitution is like finding a clever shortcut to solve integration problems. It involves changing complex expressions into simpler ones within integrals, making them easier to solve. In the given exercise, the inner part of the integrand was identified as \( u = \sqrt{x} \). By substituting \( u \), we reframe the problem, often turning it into a more manageable form.

Using substitution requires us to also change the differential \( dx \) into terms of \( du \). For instance, we derived that \( dx = 2u \, du \), allowing us to transform the original integral into a simplified form with respect to \( u \). The key ideas here include:

• Identifying a suitable substitution that simplifies the integrand.
• Expressing all components, including \( dx \), in terms of the new variable.
• Simplifying the integral which becomes easier to evaluate.

Variable substitution is a crucial skill in calculus, supporting the evaluation of otherwise challenging integrals.

Understanding Indefinite Integrals

An indefinite integral is the reverse process of differentiation; it finds a function whose derivative is the integrand.Unlike definite integrals, indefinite integrals do not have upper and lower limits, which is why they include a constant of integration, \( C \).

In our exercise, after the substitution, the integral \( 2 \int e^u \, du \) represents an indefinite integral.This means we are finding the original function before it was differentiated to \( e^u \).

• Indefinite integrals give families of functions differing by a constant.
• The constant of integration \( C \) accounts for any vertical shifts.
• It’s essential when solving initial value problems or establishing general antiderivatives.

Grasping indefinite integrals is fundamental for solving a variety of calculus problems as it opens doors to understanding how functions behave.

Decoding the Exponential Function

The exponential function, \( e^x \), is unique and incredibly powerful in calculus due to its special property.When integrated or differentiated, it remains the same, aside from any multiplicative constants.This feature makes solving integrals involving \( e^x \) straightforward compared to other functions.

In our rewritten integral from the exercise, \( 2 \int e^u \, du \), the integral directly leads to \( 2e^u \).Why is it so simple?

• Exponentials are self-replicating under differentiation and integration.
• \( \int e^x \, dx = e^x + C \), showing no need for complex manipulation.
• Exponential decay and growth models frequently use this function.

Understanding exponentials is vital, as these functions appear in math and real-world scenarios, describing everything from population growth to radioactive decay.