Problem 9 - Single Variable Calculus

Question

Find the indefinite integral, and check your answer by differentiation. $$ \int\left(e^{t}+t^{e}\right) d t $$

Step-by-Step Solution

Answer

The indefinite integral of the given function is: \[ \int\left(e^{t}+t^{e}\right) d t = e^t + \int t^e\:dt + C \] where \(C\) is the constant of integration.

Step 1: Integrate the first term, \(e^t\)

To find the indefinite integral of \(e^t\), we recall that the integral of \(e^t\) with respect to \(t\) is simply \(e^t\): \[ \int e^t\:dt = e^t + C_1 \] where \(C_1\) is the constant of integration for the first term.

Step 2: Integrate the second term, \(t^e\)

Unfortunately, no elementary anti-derivative exists for \(t^e\), so we leave it as it is for an indefinite integral: \[ \int t^e\:dt = \int t^e\:dt + C_2 \] where \(C_2\) is the constant of integration for the second term.

Step 3: Combine the results of the integration of both terms and find the indefinite integral

Adding the results obtained in Step 1 and Step 2, we get the indefinite integral: \[ \int(e^t + t^e)\:dt = \left(e^t + C_1\right) + \left(\int t^e\:dt + C_2\right) \] Combining the constants of integration, we can write the indefinite integral as: \[ \int\left(e^{t}+t^{e}\right) d t = e^t + \int t^e\:dt + C \] where \(C\) is the combined constant of integration.

Step 4: Differentiate the result to check the answer

Differentiating our result to verify the solution, we get: \[ \frac{d}{dt}\left(e^t + \int t^e\:dt + C\right) \] The derivative of \(e^t\) is \(e^t\), and the derivative of \(\int t^e\:dt\) is simply \(t^e\). The derivative of the constant \(C\) is 0. Adding the derivatives of the individual terms, we get: \[ e^t + t^e \] Since the derivative matches the original function, our indefinite integral is correct.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus. It refers to the process of finding the derivative of a function. The derivative measures how a function's output value changes as its input changes.

• The derivative of a function at a point is defined as the slope of the tangent line to the graph of the function at that point.
• Derivatives are used to find rates of change, determine the slope of a curve, and solve real-world problems involving rates.

In our exercise, we use differentiation to check the correctness of our indefinite integral. By differentiating the integrated function, we should obtain the original function that was given in the problem. If not, there could be an error in our integration.

Integration by Parts

Integration by parts is a technique used to find integrals of products of functions. It is based on the product rule for differentiation and is given by the formula:\[\int u\, dv = uv - \int v\, du\]where:- \(u\) and \(dv\) are parts of the original integral. - \(v\) and \(du\) are derived from those parts.

• This method is particularly useful when dealing with polynomial and exponential or trigonometric functions.
• It can transform an otherwise difficult integral into a simpler form that is easier to solve.

In the given exercise, integration by parts was applied implicitly as recognizing that \(e^t\) can be handled directly while leaving \(t^e\) since it does not fit into any simple integration formula.

Exponential Functions

Exponential functions are functions of the form \(f(x) = a^x\), where \(a\) is a positive constant. Exponential functions grow rapidly and have unique properties, particularly with respect to calculus:

• The derivative of an exponential function with base \(e\), \(e^x\), is \(e^x\) itself, making it unique and easy to work with during differentiation.
• The integral of \(e^x\) is also \(e^x\), plus the constant of integration, which simplifies integration tasks involving exponential functions.

In our problem, we easily integrate \(e^t\) due to its straightforward nature as an exponential function. This demonstrates how exponential functions' properties simplify calculus operations, a key learning point for tackling advanced integration problems.