Problem 25 - Calculus for Biology and Medicine

Question

Use substitution to evaluate the indefinite integrals. $$ \int 3 x e^{x^{2}} d x $$

Step-by-Step Solution

Answer

The integral evaluates to \(\frac{3}{2} e^{x^2} + C\).

Step 1: Identify Inner Function

Examine the integrand to identify a function whose derivative is also present. In this case, observe that we have \(x^2\) inside the exponential, and its derivative \(2x\) is related to the \(x\) term outside.

Step 2: Choose Substitution

Decide on a substitution to simplify the integral. Set \(u = x^2\), so that \(du = 2x \, dx\). This implies \(x \, dx = \frac{1}{2}du\).

Step 3: Rewrite the Integral

Rewrite the original integral in terms of \(u\). The original integral \(\int 3x e^{x^2} \, dx\) becomes \(\int 3 e^{u} (\frac{1}{2})du\), which simplifies to \(\frac{3}{2} \int e^{u} \, du\).

Step 4: Integrate with Respect to u

Evaluate the integral \(\int e^u \, du\). The antiderivative of \(e^u\) is \(e^u + C\). So, \(\frac{3}{2} \int e^u \, du = \frac{3}{2} e^u + C\).

Step 5: Substitute Back to x

Replace \(u\) with \(x^2\) to express the antiderivative in terms of \(x\). Thus, the solution becomes \(\frac{3}{2} e^{x^2} + C\).

Key Concepts

Substitution Method

The substitution method is an essential technique in integral calculus used to simplify and evaluate integrals. This method involves substituting a part of the integrand with a new variable, typically denoted as \( u \), to transform the integral into a simpler form. To effectively apply the substitution method:

• Select an "inner" function in the integral; often this is the more complex expression inside another function, like \( x^2 \) inside \( e^{x^2} \).
• Determine the derivative of this function; for \( x^2 \), the derivative is \( 2x \).
• Make the substitution \( u = x^2 \), then express \( du \) in terms of \( dx \), resulting in \( du = 2x \, dx \).

This substitution transforms the integral into a simpler one, often easier to evaluate. It's a key technique that helps in finding antiderivatives of functions that otherwise appear complex.

Indefinite Integrals

Indefinite integrals, often referred to simply as antiderivatives, are the inverse of differentiation. Finding an indefinite integral of a function means determining a function whose derivative matches the original function. In the context of the provided exercise, the goal was to find the indefinite integral of \( 3x e^{x^2} \). Here’s what an indefinite integral involves:

• The result contains a family of functions, commonly expressed with \( + C \) to indicate the constant of integration, as there are infinitely many antiderivatives that differ by a constant.
• The notation \( \int f(x) \, dx \) defines the indefinite integral, where \( f(x) \) is the function being integrated.

Understanding indefinite integrals is crucial since they form the basis for solving various problems in integral calculus and aid in understanding accumulated quantities over intervals.

Antiderivative

An antiderivative is a function whose derivative is a given function. It's the key result of solving indefinite integrals. In simple terms, if you differentiate an antiderivative, you should get back the original function. To comprehend antiderivatives better:

• Consider \( e^u \) in our exercise, where the derivative of the antiderivative \( e^u \) returns \( e^u \) itself.
• When integrating \( e^u \), we obtain the antiderivative \( e^u + C \). This represents the family of functions that differentiate to \( e^u \).

Finding an antiderivative is a central task in integral calculus, enabling the calculation of areas, solving differential equations, and understanding the underlying changes within functions.

Change of Variables

Change of variables is a technique related to substitution, used to simplify integrals by transforming the variable of integration. This approach changes the integral into a different, often simpler form.Psychologically, the change of variables might sound tricky, but it's manageable by following these steps:

• Identify the substitution variable \( u \) that will replace part of the original integral; here, \( u = x^2 \).
• Express \( dx \) in terms of \( du \), like from \( du = 2x \, dx \) leading to \( x \, dx = \frac{1}{2} du \).
• Rewrite the integral with the new variable, transforming the original integral into a more straightforward problem to solve.
• After integrating, translate back to the initial variable to present the solution in its original context.

This method, essential in calculus, facilitates dealing with complex integrals by transforming them into ones that are easier to integrate. This is invaluable in solving real-world problems involving rates of change and accumulation.