Question

a. Use integration by parts to show that if \(f^{\prime}\) is continuous, $$\int x f^{\prime}(x) d x=x f(x)-\int f(x) d x$$ b. Use part (a) to evaluate \(\int x e^{3 x} d x\).

Short answer

Question: Using integration by parts, show that for any function \(f\) whose derivative \(f'\) is continuous, the integral \(\int x f'(x)dx = x f(x) - \int f(x)dx\). Then, evaluate the integral \(\int xe^{3x}dx\). Answer: The integration by parts technique shows that \(\int x f'(x)dx = x f(x) - \int f(x)dx\), as demonstrated in the solution above. Using this formula, we evaluate the given integral \(\int xe^{3x}dx\) as follows: $$\int xe^{3x} dx = \frac{1}{3}xe^{3x} - \frac{1}{9}e^{3x} + C.$$

Step-by-step solution

Integration by Parts Formula

Recall the integration by parts formula: \(\int u~dv = uv - \int v~du\). Choose \(u=x\) and \(dv=f^{\prime}(x)dx\) for our given integral. Now, we need to find \(du\) and \(v\).

Calculate du and v

For \(u=x\), we have \(du = dx\). Since \(dv = f^{\prime}(x)dx\), then the integration of \(dv\) gives us \(v = f(x)\).

Apply the Integration by Parts Formula

Now, apply the integration by parts formula to our expressions for \(u\), \(v\), \(du\), and \(dv\): $$\int x f^{\prime}(x) dx = x f(x) - \int f(x) dx.$$ Thus, we have shown the desired formula. Now, let's move to part (b) where we will use the derived formula to evaluate the given integral.

Identify the Function and its Derivative

In this case, we have \(\int xe^{3x} dx\). Here, we will identify \(f^{\prime}(x) = e^{3x}\), hence the function \(f(x)\) would be its antiderivative.

Calculate the Antiderivative of the Exponential Function

To find the antiderivative \(f(x)\), we will calculate the integral of \(f^{\prime}(x) = e^{3x}\): $$f(x) = \int e^{3x} dx = \frac{1}{3}e^{3x} + C,$$ Where \(C\) is the constant of integration.

Apply the Established Formula

Now, we apply the formula we derived in part (a): $$\int x e^{3x} dx = x f(x) - \int f(x) dx = x\left(\frac{1}{3}e^{3x}\right) - \int \left(\frac{1}{3}e^{3x}\right) dx.$$

Calculate the Final Integral

Finally, we need to find the integral of the resulting expression: $$\int x e^{3x} dx = x\left(\frac{1}{3}e^{3x}\right) - \frac{1}{9}e^{3x} + C = \frac{1}{3}xe^{3x} - \frac{1}{9}e^{3x} + C.$$

Key concepts

Antiderivative

An antiderivative of a function is essentially an inverse operation to differentiation. In other words, it is a function whose derivative is the original function we started with.For example, if we have a function whose derivative is given by some expression, then the antiderivative is the function itself before it was differentiated. The process of finding an antiderivative is called "integration."
When you find an antiderivative, it usually includes a constant of integration, represented as "C." This is because differentiation of a constant is zero, and hence it can be any real number. For instance, if the derivative of a function is a constant, like 3, then its antiderivative will include an arbitrary constant, such as:

• The antiderivative of 3 is: 3x + C

In integration by parts, we often need to compute the antiderivative of part of the integrand, as seen in the step where we find the integral of the exponential function, like for the function \(f^{ ext{'}(x)} = e^{3x}\).
Antiderivatives form the basis of both definite and indefinite integrals, as integration refers to calculating antiderivatives to solve multiple types of problems.

Exponential Functions

Exponential functions are powerful mathematical expressions that involve powers of a constant base raised to a variable exponent, typically represented as \(e^{x}\) where \(e\) is Euler's constant, approximately equal to 2.71828.
Exponential growth and decay are common phenomena observed in nature and economics. In mathematics, these functions have unique properties, such as their derivative being proportional to the function itself. This can be noted as:

• The derivative \(\frac{d}{dx} e^{3x} = 3e^{3x}\)
• The same form holds for any constant multiple within the exponent

In the context of integration, computing the antiderivative of an exponential function often appears in problems. For instance, to find the antiderivative of \(e^{3x}\), we perform integration and adjust for the chain rule applied during differentiation by dividing by the constant in the exponent, yielding the solution:\[\int e^{3x} dx = \frac{1}{3} e^{3x} + C\]
Mastering exponential functions is critical in solving calculus problems, especially when using methods like integration by parts, as they frequently appear in mathematically modeling real-world problems.

Definite and Indefinite Integrals

Integrals in calculus can mainly be classified as either definite or indefinite.An indefinite integral, often referred to simply as an "antiderivative," involves finding a general formula for the antiderivative of a function without specific bounds. It is expressed in the form:

• \(\int f(x) \, dx = F(x) + C\)

where \(F(x)\) is any antiderivative of \(f(x)\), and \(C\) is the constant of integration.
On the other hand, definite integrals calculate the net area under a curve between two bounds, \(a\) and \(b\), and are given by:

• \(\int_{a}^{b} f(x) \, dx = F(b) - F(a)\)

The bounds \(a\) and \(b\) constitute the limits of integration, and the result is a number representing the total accumulation between these points.
In the exercise, when applying integration by parts to evaluate the integral \(\int x e^{3x} dx\), we're demonstrating the use of indefinite integrals. It involves integrating without specific bounds, illuminating the process of finding an antiderivative useful for a variety of problems. Understanding both definite and indefinite integrals is fundamental to fully grasping integral calculus and its practical applications.