Question

$$ \text { use integration by parts to evaluate each integral. } $$ $$ \int x e^{3 x} d x $$

Short answer

\( \frac{x}{3} e^{3x} - \frac{1}{9} e^{3x} + C \)

Step-by-step solution

Identify Parts of the Formula

For integration by parts, use the formula \( \int u \, dv = uv - \int v \, du \). Choose \( u = x \) (since its derivative becomes simpler when differentiated) and \( dv = e^{3x} \, dx \).

Differentiate and Integrate

Compute \( du \) by differentiating \( u \): \( du = dx \). Compute \( v \) by integrating \( dv \): \( v = \int e^{3x} \, dx = \frac{1}{3} e^{3x} \).

Apply Integration by Parts Formula

Substitute into the integration by parts formula: \( \int x e^{3x} \, dx = x \cdot \frac{1}{3} e^{3x} - \int \frac{1}{3} e^{3x} \, dx \).

Simplify the Expression

Simplifying gives \( \frac{x}{3} e^{3x} - \frac{1}{3} \int e^{3x} \, dx \).

Solve Remaining Integral

Evaluate the integral \( \int e^{3x} \, dx \), which is \( \frac{1}{3} e^{3x} \), to get \( \frac{x}{3} e^{3x} - \frac{1}{9} e^{3x} \).

Combine Terms and Add Constant

Combine the terms: \( \frac{x}{3} e^{3x} - \frac{1}{9} e^{3x} + C \), where \( C \) is the constant of integration.

Key concepts

Definite Integral

A definite integral is a fundamental concept in calculus, used to calculate the area under a curve between two specified points on the x-axis. It gives a precise numerical value, unlike the indefinite integral. The notation for definite integrals is written as:

• \( \int_{a}^{b} f(x) \ dx \)

where \( a \) and \( b \) are the lower and upper limits of integration, respectively. This notation tells us to evaluate the antiderivative of \( f(x) \) from \( a \) to \( b \).
To solve a definite integral during computation, follow these steps:

• Find the indefinite integral of the function.
• Calculate the values of this antiderivative at both bounds \( a \) and \( b \).
• Subtract the antiderivative value at \( a \) from the value at \( b \).

This process gives the total accumulated value from point \( a \) to point \( b \). A practical example might be evaluating the area under the function \( f(x) = x^2 \) from 1 to 3. The definite integral results in a specific number, representing this area under the curve.

Indefinite Integral

The indefinite integral, often represented without limits, is a standard way to find the family of functions that represent antiderivatives of a given function. It contains an arbitrary constant denoted as \( C \) because of this family of solutions.
The notation for indefinite integrals is:

• \( \int f(x) \, dx = F(x) + C \)

where \( F(x) \) is the antiderivative, and \( C \) represents any constant since differentiating a constant results in zero.
In practical terms, the indefinite integral of a function gives us the general form of all possible antiderivatives for that function. For instance, when computing \( \int x \, dx \), we get \( \frac{1}{2} x^2 + C \), representing the family of parabolas shifted by any amount vertically. This approach encapsulates the inherent flexibility in calculus when deriving functions from their rates of change.

Calculus Techniques

Calculus techniques are essential tools for solving integrals and derivatives in a variety of complex scenarios. One such technique, **integration by parts**, is especially useful when dealing with the product of functions:

• The integration by parts formula is derived from the product rule of differentiation and stated as:\( \int u \, dv = uv - \int v \, du \).

This method involves:

• Choosing parts of your function as \( u \) and \( dv \) strategically.
• Differentiating \( u \) and integrating \( dv \).
• Applying the integration by parts formula to solve the integral.

For example, in the integral \( \int x e^{3x} \, dx \), appropriate choices and applications of differentiation and integration simplify this complex expression into manageable parts. Additionally, other techniques like substitution, partial fraction decomposition, and trigonometric integrals are pivotal in analyzing and solving integrals.
Such methods reveal the power and flexibility of calculus, allowing for solutions of various function types and complexity levels.