Problem 88 - Calculus Single Variable

Question

The error function (denoted by erf) is central to the subjects of probability and statistics. It is defined by $$ \operatorname{erf}(x)=\frac{2}{\sqrt{\pi}} \int_{0}^{x} \exp \left(-t^{2}\right) d t $$ a. Integrate by parts to calculate \(\int \operatorname{erf}(x) d x\). (Your answer will involve erf \((x)\) and elementary functions.) b. Integrate by parts to calculate \(\int x^{2} \exp \left(-x^{2}\right) d x .\) (Take \(u=x, d v=x \exp \left(-x^{2}\right) d x .\) Your answer will involve erf \((x)\) and elementary functions.) c. Integrate by parts to calculate \(\int x^{2} \operatorname{erf}(x) d x\). (You will need the result of part b. Your answer will involve \(\operatorname{erf}(x)\) and elementary functions.)

Step-by-Step Solution

Answer

1a. \(x\operatorname{erf}(x) + \frac{1}{\sqrt{\pi}} e^{-x^2} + C\); 1b. \(-\frac{1}{2} x e^{-x^2} + \frac{\sqrt{\pi}}{4}\operatorname{erf}(x) + C\); 1c. \(x^3\operatorname{erf}(x) + \frac{x^2}{\sqrt{\pi}} e^{-x^2} + x e^{-x^2} - \frac{\sqrt{\pi}}{2} x \operatorname{erf}(x) + C\)."

Step 1: Setup Integration by Parts for erf(x)

First, we recall the integration by parts formula: \[\int u \, dv = uv - \int v \, du.\]For \(\int \operatorname{erf}(x) \, dx\), choose \(u = \operatorname{erf}(x)\) and \(dv = dx\), thus \(du = \frac{2}{\sqrt{\pi}} e^{-x^2} \; dx\) and \(v = x\).

Step 2: Apply Integration by Parts for erf(x)

Now, apply the integration by parts formula:\[\int \operatorname{erf}(x) \, dx = x \operatorname{erf}(x) - \int x \cdot \frac{2}{\sqrt{\pi}} e^{-x^2} \, dx.\]Simplify the resulting integral for further calculation.

Step 3: Simplify the Result for Part a

The first integral part gives \(x \operatorname{erf}(x)\). For the second integral, rewrite it:\[-\frac{2}{\sqrt{\pi}} \int x e^{-x^2} \, dx,\]which is a standard integral, and its solution is: \[-\frac{1}{\sqrt{\pi}} e^{-x^2} + C.\]Therefore, \[\int \operatorname{erf}(x) \, dx = x \operatorname{erf}(x) + \frac{1}{\sqrt{\pi}} e^{-x^2} + C.\]

Step 4: Setup Integration by Parts for Part b

Now, for \(\int x^2 e^{-x^2} \, dx\), choose \(u = x\) and \(dv = x e^{-x^2} \, dx\). Thus, \(du = dx\) and \[v = -\frac{1}{2} e^{-x^2}.\]

Step 5: Apply Integration by Parts for Part b

Using integration by parts:\[\int x^2 e^{-x^2} \, dx = -\frac{1}{2} x e^{-x^2} - \int -\frac{1}{2} e^{-x^2} \, dx x,\]Simplifying:\[= -\frac{1}{2} x e^{-x^2} + \frac{1}{2} \int e^{-x^2} \, dx.\]The second integral can be expressed in terms of the error function: \[= -\frac{1}{2} x e^{-x^2} + \frac{\sqrt{\pi}}{4} \operatorname{erf}(x) + C.\]

Step 6: Setup Integration by Parts for Part c

For part c, \(\int x^2 \operatorname{erf}(x) \, dx\), choose \(u = x^2\) and \(dv = \operatorname{erf}(x) \, dx\). Therefore, \(du = 2x \, dx\) and the previous integration by parts result applies: \[v = x \operatorname{erf}(x) + \frac{1}{\sqrt{\pi}} e^{-x^2} + C.\]

Step 7: Apply Integration by Parts for Part c

Using integration by parts:\[\int x^2 \operatorname{erf}(x) \, dx = x^2 \left( x \operatorname{erf}(x) + \frac{1}{\sqrt{\pi}} e^{-x^2} \right) - \int 2x \left( x \operatorname{erf}(x) + \frac{1}{\sqrt{\pi}} e^{-x^2} \right) \, dx.\]This breaks into:\[x^3 \operatorname{erf}(x) + \frac{x^2}{\sqrt{\pi}} e^{-x^2} - \int 2x^2 \operatorname{erf}(x) \, dx - \frac{2}{\sqrt{\pi}} \int x e^{-x^2} \, dx.\]

Step 8: Simplify Result for Part c

Simplify further using results from earlier integrals:\[= x^3 \operatorname{erf}(x) + \frac{x^2}{\sqrt{\pi}} e^{-x^2} - 2 \int ( -\frac{1}{2} x e^{-x^2} + \frac{\sqrt{\pi}}{4} \operatorname{erf}(x) ) \, dx \]Continuing with simplification provides:\[x^3 \operatorname{erf}(x) + \frac{x^2}{\sqrt{\pi}} e^{-x^2} + x e^{-x^2} - \frac{\sqrt{\pi}}{2} x \operatorname{erf}(x) + C.\]

Key Concepts

Error Function

The error function, commonly denoted as \( \operatorname{erf}(x) \), appears frequently in probability and statistics. It is primarily used to express the probability of random variables within a normal distribution. The function is defined as an integral:\[\operatorname{erf}(x) = \frac{2}{\sqrt{\pi}} \int_0^x e^{-t^2} \, dt\]This integral computes the area under the curve of the Gaussian function \( e^{-t^2} \) from 0 to \( x \), scaled by the factor \( \frac{2}{\sqrt{\pi}} \). Here’s why it’s important:

• **Probability Interpretation:** The error function is used to calculate cumulative distribution functions for normal distributions.
• **Connection to Normal Distribution:** It is closely related to the complementary error function, which helps in representing the "tail" probability of a normal distribution.

When tackling problems with the error function, such as in integration by parts, it is essential to set it properly as either \( u \) or \( dv \) depending on the problem to simplify the integration.

Definite Integrals

Definite integrals compute the exact area under the curve of a function between two specified bounds. This is in contrast to indefinite integrals, which represent a family of functions. In the context of the error function, the definite integral is used to define \( \operatorname{erf}(x) \).To solve a definite integral, follow these general steps:

• Identify the bounds of integration, which are typically given by the limits \( a \) and \( b \).
• Find the antiderivative of the integrand, the function being integrated.
• Evaluate this antiderivative at the upper and lower limits of the integral, performing a subtraction to find the area.

One should remember that definite integrals such as those involved in the error function necessitate careful attention to limits and potential changes of variables. They form the backbone of many calculations in probability, helping to determine probabilities and expected values.

Exponential Functions

Exponential functions are crucial in calculus, especially when dealing with integration problems similar to those found in error functions. These functions often appear in forms such as \( e^{x} \) or \( e^{-x^2} \), as seen in the integrals involving \( \operatorname{erf}(x) \).Key characteristics of exponential functions:

• **Growth and Decay:** They model processes of growth and decay, which are applicable in fields like biology, chemistry, physics, and economics.
• **Integration and Differentiation:** The derivative of an exponential function is proportional to the function itself, aiding in simplification during calculus operations.

When integrated, mathematics involving exponential functions and terms like \( e^{-x^2} \) can often use substitution or integration by parts for simplification. Their occurrence in problems require a deep understanding of these properties to manipulate them adequately during calculations.