Problem 51 - Calculus Single Variable

Question

Use the Chain Rule to calculate the given indefinite integral. $$ \int x \exp \left(x^{2}\right) d x $$

Step-by-Step Solution

Answer

\( \frac{1}{2} e^{x^2} + C \)

Step 1: Identify the Inner Function

First, identify the function inside the exponential which will be used as substitution. In this case, the inner function is \( u = x^2 \).

Step 2: Differentiate the Inner Function

Compute the derivative of the inner function \( u = x^2 \). We have \( \frac{du}{dx} = 2x \). Therefore, \( du = 2x \, dx \).

Step 3: Create Substitution for dx

From the derivative \( du = 2x \, dx \), solve for \( dx \). This gives \( dx = \frac{du}{2x} \).

Step 4: Substitute into the Integral

Replace \( x^2 \) with \( u \) and \( dx \) with \( \frac{du}{2x} \) in the integral. The integral becomes: \[ \int x e^{x^2} dx = \int x e^{u} \frac{du}{2x} = \frac{1}{2} \int e^{u} du \].

Step 5: Integrate with Respect to u

Now integrate \( \frac{1}{2} \int e^{u} du \). The integral of \( e^u \) with respect to \( u \) is \( e^u \). Thus, \[ \frac{1}{2} \int e^{u} du = \frac{1}{2} e^u + C \] where \( C \) is the constant of integration.

Step 6: Substitute Back for x

Replace \( u \) back with \( x^2 \) since \( u = x^2 \). Therefore, the integral becomes: \( \frac{1}{2} e^{x^2} + C \).

Key Concepts

Integration by Substitution

Integration by substitution is a fundamental technique used to simplify the process of finding antiderivatives. It's particularly useful when dealing with composite functions, where one function is nested within another, just like the problem of integrating \( \int x \exp(x^2) \, dx \). The main goal of substitution is to rewrite an integral in terms of a new variable, making it easier to solve.

Here's how it generally works:

• Identify the inner function: Recognize a part of the integral that's complicated or nested, known as the "inner function." In our example, the inner function is \( u = x^2 \).
• Differentiate the inner function: Find its derivative, \( \frac{du}{dx} \), and express \( du \) in terms of \( dx \) and any other terms necessary. For \( u = x^2 \), we have \( du = 2x \, dx \).
• Substitute: Replace the identified parts in the integral, transforming it into a simpler form. For instance, replace \( dx \) with \( \frac{du}{2x} \) and \( x^2 \) with \( u \).
• Simplify and integrate: Perform the integration with the new variable. In our case, it simplifies to \( \frac{1}{2} \int e^u \, du \).
• Back-substitute: Once the integration is complete, substitute back the original variable. Replace \( u \) with \( x^2 \) to get the final antiderivative.

This technique transforms difficult integrals into more manageable problems, utilizing the relationship between functions and their derivatives.

Indefinite Integral

An indefinite integral represents the collection of all antiderivatives of a function, depicted as \( \int f(x) \, dx \). Unlike definite integrals that compute a specific numerical value, indefinite integrals yield a general expression plus a constant, often denoted by \( C \). This constant emerges because derivatives of constant terms are zero, hence they vanish during differentiation.

In the case of \( \int x \exp(x^2) \, dx \), the expression \( \frac{1}{2} e^{x^2} + C \) is an indefinite integral. It suggests that there are infinitely many functions that could derive to \( x \exp(x^2) \), differing by a constant. This is crucial because indefinite integrals reflect a family of functions rather than a unique solution.

Remember that when solving indefinite integrals, integration techniques like substitution are often required to simplify the problem into solvable forms. The integration process itself essentially reverses differentiation, uncovering the original function which, when differentiated, would yield the integrand.

Exponential Function

Exponential functions are powerful mathematical tools characterized by their constant rate of growth or decay, expressed as \( y = a e^{bx} \), where \( e \approx 2.71828 \) is Euler's number. They are central in calculus and applications across sciences because of their unique derivative and integral properties. The exponential function is rather special because the derivative and the integral of \( e^x \) are simply \( e^x \), preserving the function's form.

In calculus, exponential functions often appear within complex integrals, like \( \int x \exp(x^2) \, dx \). Here, recognizing the exponential part of the function is critical. The standard form \( e^x \) or transformations like \( e^{g(x)} \), where \( g(x) \) is another function, allows for these uses.

Solving integrals featuring exponential functions frequently involves substitution, making it easier to treat expressions involving composite functions such as \( x^2 \) under the exponent. Ultimately, mastering the behavior and manipulation of exponential functions is vital to effectively approaching a wide variety of calculus problems.