Problem 8 - Calculus 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

\(\int e^{x} d x=e^{x}+c\)

Step-by-Step Solution

Answer

The integral of \( e^x \) is \( e^x + C \).

Step 1: Identify the Function

The given problem is asking for the integral of the function \( e^{x} \) with respect to \( x \). The function to be integrated is \( e^x \).

Step 2: Recall Integration Rule for Exponential Function

The integral of the exponential function \( e^x \) is one of the basic rules in calculus: \( \int e^{x} \, dx = e^x + C \). Here, \( C \) is the constant of integration.

Step 3: Apply the Rule to the Given Problem

Using the integration rule for exponential functions, we directly write the solution as \( \int e^{x} \, dx = e^{x} + C \).

Step 4: Finalize the Solution

The integration of \( e^x \) results in adding \( C \), the constant of integration, because indefinite integrals account for any constant added after differentiation. Therefore, the solution is \( e^{x} + C \).

Key Concepts

Exponential Function

An exponential function is a mathematical expression in which a constant base is raised to a variable exponent. In the context of calculus, one of the most common exponential functions is \( e^x \), where \( e \) is the Euler's number, approximately 2.71828. Euler's number \( e \) has unique properties that make \( e^x \) particularly interesting:

• The rate at which \( e^x \) increases is proportional to its current value. This is why it appears regularly in growth and decay problems.
• It is the only function whose derivative and integral are exactly itself.

This aspect makes \( e^x \) fundamental in calculus, especially when dealing with continuous growth processes, such as population growth or compound interest.

Constant of Integration

In integration, especially when dealing with indefinite integrals, the constant of integration \( C \) plays a crucial role. When taking the indefinite integral of a function, such as \( e^x \), the result is a family of functions that differ by a constant value. This is because the derivative of any constant is zero.

• The constant of integration \( C \) represents these possible shifts in the function's vertical position on a graph.
• In practical terms, \( C \) accounts for initial conditions or specific scenarios in applied problems where the exact value of the function at a particular point is known.

Thus, after integrating a function like \( e^x \), you write the result as \( e^x + C \) to signify this variety of possible solutions.

Indefinite Integrals

Indefinite integrals refer to the integration of a function without specified limits. Unlike definite integrals that compute a number representing the area under a curve within a given range, indefinite integrals result in a function.

• The notation \( \int f(x) \, dx \), where no upper and lower bounds on the integral sign appear, denotes an indefinite integral.
• The result is a family of functions that includes the constant of integration \( C \), as no specific boundaries restrict the solution.

The purpose of finding an indefinite integral is often to "reverse" differentiation. For instance, if you started with a function \( e^x \) and took its derivative, you would end up with \( e^x \) again. To return to the original function before differentiation, you perform integration, resulting in \( e^x + C \). This process underscores the relationship between differentiation and integration, two fundamental operations in calculus.