Problem 76 - Applied Calculus

Question

Sometimes an integral requires two or more integrations by parts. As an example, we apply integration by parts to the integral \(\int x^{2} e^{x} d x\). \(\int \underbrace{x^{2} e^{x} d x}=x^{2} e^{x}-\int_{u} \underbrace{e^{x} 2 x d x}=x^{2} e^{x}-2 \int x e^{x} d x\) \(\left[\begin{array}{cc}u=x^{2} & d v=e^{x} d x \\ d u=2 x d x & v=\int e^{x} d x=e^{x}\end{array}\right]\) The new integral \(\int x e^{x} d x\) is solved by a second integration by parts. Continuing with the previous solution, we choose new \(u\) and \(d u\) : \(=x^{2} e^{x}-2\left(\int x e^{x} d x\right) \quad\left[\begin{array}{c}u=x \quad d v=e^{x} d x \\ d u=d x \quad v=e^{x}\end{array}\right]\) \(=x^{2} e^{x}-2\left(x e^{x}-\int e^{x} d x\right)\) \(=x^{2} e^{x}-2\left(x e^{x}-e^{x}\right)+C\) \(=x^{2} e^{x}-2 x e^{x}+2 e^{x}+C\) After reading the preceding explanation, find each integral by repeated integration by parts. \(\int x^{2} e^{2 x} d x\)

Step-by-Step Solution

Answer

\( \int x^2 e^{2x} \, dx = \frac{1}{2} x^2 e^{2x} - \frac{1}{2} x e^{2x} + \frac{1}{4} e^{2x} + C \).

Step 1: Identify Parts for Integration by Parts

For the integral \( \int x^2 e^{2x} \, dx \), apply the integration by parts formula, \( \int u \, dv = uv - \int v \, du \). Choose \( u = x^2 \) and \( dv = e^{2x} \, dx \).

Step 2: Differentiate and Integrate

Differentiate \( u = x^2 \) to get \( du = 2x \, dx \). Integrate \( dv = e^{2x} \, dx \) to get \( v = \frac{1}{2}e^{2x} \).

Step 3: Apply Integration by Parts Formula

Substitute into the integration by parts formula: \( \int x^2 e^{2x} \, dx = x^2 \left( \frac{1}{2} e^{2x} \right) - \int \left( \frac{1}{2}e^{2x} \right) \cdot 2x \, dx \). Simplify this to obtain: \( \frac{1}{2} x^2 e^{2x} - \int x e^{2x} \, dx \).

Step 4: Second Integration by Parts

The integral \( \int x e^{2x} \, dx \) requires another integration by parts. Choose \( u = x \) and \( dv = e^{2x} \, dx \). Differentiate \( u = x \) to get \( du = dx \), and integrate \( dv = e^{2x} \, dx \) to get \( v = \frac{1}{2} e^{2x} \).

Step 5: Substitute Second Integration by Parts

Substitute the second set of parts into the integration by parts formula: \( \int x e^{2x} \, dx = x \cdot \frac{1}{2} e^{2x} - \int \frac{1}{2} e^{2x} \, dx \). This simplifies to \( \frac{1}{2} x e^{2x} - \frac{1}{4} e^{2x} + C \).

Step 6: Combine Results

Substitute the result of the second integration back: \( \frac{1}{2} x^2 e^{2x} - \left( \frac{1}{2} x e^{2x} - \frac{1}{4} e^{2x} \right) + C \). Simplify to \( \frac{1}{2} x^2 e^{2x} - \frac{1}{2} x e^{2x} + \frac{1}{4} e^{2x} + C \).

Key Concepts

Definite Integrals

A definite integral is a way to find the area under a curve within certain limits. This means you're looking from one specific point to another on a graph. It tells you how much space is beneath a graph from one point to another.

Definite integrals have specific properties:

• They have limits, shown as the numbers at the top and bottom of the integral sign. These limits tell you where to start and stop on the x-axis.
• The result of a definite integral is a real number. It gives a cumulative net change over an interval.
• They are often used in physics and engineering to calculate things like area, volume, and total accumulated change.

Applying this to integration by parts, while the method focuses on indefinite integrals, you can use the results within a definite integral to evaluate areas between curves or points.

Antiderivatives

Antiderivatives are the opposite of derivatives. They tell you what function was differentiated to get a particular derivative.In simpler terms, if a derivative is a slope or a rate of change, an antiderivative takes that change and finds the original function it came from. This process is called integration.

When using integration by parts:

• Choose which part of the function to differentiate (choice of 'u') and which to integrate (choice of 'dv').
• Apply the formula: \( \int u \, dv = uv - \int v \, du \). This helps in obtaining the antiderivative.
• Sometimes, multiple steps are required, involving more than one integration by parts, to find the complete antiderivative.

By understanding antiderivatives, you can solve integrals and work backwards from rates of change to original quantities.

Exponential Functions

Exponential functions are a type of mathematical function that involve expressions with variables as exponents, like \( e^{x} \). They have distinctive properties:

• They grow at an ever-increasing rate. As the variable increases, so does the function's value, often very rapidly.
• The base \( e \) is a special constant approximately equal to 2.718. It is known as Euler's number.
• Exponential functions are used to model real-world phenomena like population growth, radioactive decay, and compound interest.

When integrating exponential functions, particularly with integration by parts, they often simplify cleanly. For example, when integrating terms like \( e^{2x} \), the integral is \( \frac{1}{2}e^{2x} \), which easily plugs back into further calculations.

Understanding exponential functions is crucial as they appear frequently in natural processes and mathematical solutions.