Problem 5 - Applied Calculus

Question

Find the integrals in problems. Check your answers by differentiation. $$ \int 2 q e^{q^{2}+1} d q $$

Step-by-Step Solution

Answer

The integral of \( \int 2q e^{q^2 + 1} \, dq \) is \( e^{q^2 + 1} + C \).

Step 1: Identify the Integral Form

The given integral is \( \int 2q e^{q^2 + 1} \, dq \). To evaluate this, we need to identify if it's in a standard form or requires substitution.

Step 2: Apply Substitution Method

Use substitution by setting \( u = q^2 + 1 \). Hence, \( du = 2q \, dq \). This simplifies the integral to \( \int e^u \, du \).

Step 3: Integrate with Respect to New Variable

The integral \( \int e^u \, du \) is straightforward and results in \( e^u + C \), where \( C \) is the constant of integration.

Step 4: Substitute Back Original Variables

Substitute back the original variable by replacing \( u \) with \( q^2 + 1 \), resulting in \( e^{q^2 + 1} + C \).

Step 5: Differentiate the Result to Verify

Differentiate \( e^{q^2 + 1} + C \) with respect to \( q \) to verify the result. By the chain rule, the derivative is \( 2q e^{q^2 + 1} \), which matches the original integrand.

Key Concepts

Substitution Method

The substitution method, often known as "u-substitution," is a powerful tool used in calculus to simplify integrals that are not in an immediately recognizable form. The idea is to make a substitution that transforms a complicated integral into an easier one, allowing for straightforward integration.
To apply the substitution method, follow these general steps:

• Identify the part of the integral to substitute, typically something that complicates the expression, like a composite function.
• Substitute this part with a new variable, usually symbolized by "u". This involves expressing the differential form as well, converting the entire differential format to "du".
• Perform the integration with respect to the new variable "u".
• Return to the original variable by substituting "u" back with its equivalent expression.

In our specific exercise, we simplified \( \int 2q e^{q^2 + 1} \, dq \) by letting \( u = q^2 + 1 \), where \( du = 2q \, dq \). This turned our complex integral into a simple exponential integral \( \int e^u \, du \). Substitution makes such problems easier by reducing the complexity involved in directly integrating the original function.

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate at which a quantity changes. It serves as the reverse process to integration and is instrumental in verifying solutions to integrals.
In our problem, after solving the integral using the substitution method, we differentiate the resulting expression to ensure the solution is correct. Here's the process:

• After integrating and returning to the variable of interest, you obtain a function, in our case, \( e^{q^2 + 1} + C \).
• Differentiate this function with respect to the original variable. For the function \( e^{q^2 + 1} \), use the chain rule. The derivative is \( 2qe^{q^2 + 1} \).
• Check if the derivative obtained matches the original function you integrated. If it does, this confirms the integration was done correctly.

Differentiation not only aids in verifying integrals but also gives insights into the behavior of functions, like finding maxima, minima, and points of inflection.

Definite and Indefinite Integrals

In calculus, integrals are classified as "definite" and "indefinite," each serving different purposes. An indefinite integral does not have limits of integration and results in a family of functions, all differing by a constant \( C \). This constant appears because antidifferentiation reverses differentiation, where constants vanish.
Definite integrals, however, calculate the net area under the curve of a function within specified boundaries. This involves evaluating the antiderivative at the upper and lower limits and finding the difference.

• Indefinite Integrals: Represented as \( \int f(x) \, dx = F(x) + C \), where \( F'(x) = f(x) \).
• Definite Integrals: Represented as \( \int_a^b f(x) \, dx = F(b) - F(a) \).

In our exercise, the integral \( \int 2q e^{q^2 + 1} \, dq \) is indefinite. Therefore, the solution is expressed as \( e^{q^2 + 1} + C \). Understanding the role of definite and indefinite integrals is crucial in fields like physics, engineering, and probability, where they are used to calculate areas, volumes, and probability distributions.