Problem 77 - 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. \(\int(x+1)^{2} e^{x} d x\)

Step-by-Step Solution

Answer

The integral \( \int(x+1)^2 e^x \, dx \) simplifies to \( x^2 e^x + 2e^x + C \).

Step 1: Recognize the Structure

To solve the integral \( \int (x+1)^2 e^x \, dx \) using integration by parts, first remember the integration by parts formula: \[ \int u \, dv = uv - \int v \, du \]Our goal is to identify the parts \( u \) and \( dv \) from the integrand \((x+1)^2 e^x \).

Step 2: First Integration by Parts Setup

Choose \( u = (x+1)^2 \) and \( dv = e^x \, dx \). Three differentiations are needed:- \( du = 2(x+1) \, dx \), and- Integrate \( dv \) to find \( v = e^x \).Apply the integration by parts:\[ \int (x+1)^2 e^x \, dx = (x+1)^2 e^x - \int e^x \, 2(x+1) \, dx \]

Step 3: Simplify and Repeat Integration by Parts

Now, simplify and set up another integration by parts for \( \int e^x \, 2(x+1) \, dx \):\[ = (x+1)^2 e^x - 2 \int (x+1) e^x \, dx \]Choose new parts: \( u = (x+1) \) and \( dv = e^x \, dx \).- \( du = dx \)- \( v = e^x \)Apply the integration by parts:\[ \int (x+1) e^x \, dx = (x+1) e^x - \int e^x \, dx \]

Step 4: Simplify the Remaining Integral

Substitute back into the equation and simplify:\[ (x+1)^2 e^x - 2 \left( (x+1) e^x - \int e^x \, dx \right) \]For the remaining integral \( \int e^x \, dx \), we know it's a base integral:\[ \int e^x \, dx = e^x + C \]

Step 5: Final Simplification

Combine all parts together and simplify completely:\[ (x+1)^2 e^x - 2 \left( (x+1) e^x - e^x \right) + C \]Expand and simplify:\[ = (x+1)^2 e^x - 2(x e^x + e^x - e^x) + C \]Thus:\[ = (x+1)^2 e^x - 2x e^x \]

Step 6: Finalize and Clean Up the Expression

Combine terms and distribute correctly to reach final simplification:\[ = (x^2 + 2x + 1)e^x - 2xe^x + 2e^x + C \]Simplified:\[ = x^2 e^x + 2e^x + C \]

Key Concepts

Repeated Integration

Repeated integration by parts is a method applied when one application of integration by parts is not enough to fully solve an integral. The idea is to apply the method iteratively until a solvable integral is reached. This technique is useful for integrals involving products of polynomial and exponential functions, as it often simplifies the problem progressively.

• Start by choosing the function that simplifies the most upon differentiation as your first part, usually denoted as \( u \).
• The remainder of the product becomes \( dv \), which is integrated to find \( v \).
• Compute new integrals using the formula: \( \int u \, dv = uv - \int v \, du \).

By repeating the integration by parts, the integrals usually simplify into a straightforward form. When faced with another complex term, continue applying the technique until reaching a final result or simple integral form, such as a standard exponential or polynomial integral.

Exponential Integration

Exponential integration deals with integrals where the integrand contains an exponential function, such as \( e^x \). These types of integrals often occur in integration by parts problems due to their simple derivative and integral properties.

• The derivative of \( e^x \) is also \( e^x \), which makes it a natural choice for the \( dv \) term in integration by parts.
• The integral of \( e^x \) simplifies to itself, \( e^x + C \), which can streamline repetitive steps in solving and helps in simplifying calculations.
• Exponential functions retain their form after differentiation and integration, which can lead to elegant and predictable solutions especially when combined with polynomial functions.

When used in combination with other functions, the consistent nature of exponential functions is a powerful asset in performing calculations efficiently and reaching a solution.

Polynomial Integration

Polynomial integration focuses on integrals involving polynomials, characterized by terms like \( x^n \). In these expressions, using integration by parts can be helpful especially when a polynomial is multiplied by a more complex function.

• Polynomials, when differentiated, reduce in degree. This property makes them ideal candidates for the \( u \) term during integration by parts.
• Successive differentiations gradually simplify the polynomial, often transforming the integral into a manageable expression or into simpler exponential-integral combinations.
• When the polynomial is an isolated integral, you can directly apply the power rule for integration: \( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \), simplifying the problem without needing integration by parts.

When managing integrals of polynomial functions, understanding how these expressions break down step-by-step creates clarity and simplicity in solving integration problems effectively.