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Chapter 7: Problem 29

Find the integrals. $$\int x^{3} e^{x^{2}} d x$$

Short Answer

Expert verified
The integral of \( x^3 e^{x^2} \, dx \) is \( \frac{1}{2} x^2 e^{x^2} - \frac{1}{2} e^{x^2} + C \).

Step by step solution

01

Identify the Integral Type

The integral \( \int x^3 e^{x^2} \, dx \) involves both a polynomial \( x^3 \) and an exponential function \( e^{x^2} \). This suggests that integration by parts might be a suitable technique, due to the product of functions.
02

Use Integration by Parts

Recall the integration by parts formula: \( \int u \, dv = uv - \int v \, du \). Choose \( u = x^2 \) (since its derivative simplifies the integral) and \( dv = x e^{x^2} \, dx \).
03

Differentiate and Integrate Parts

Differentiate \( u \): \( du = 2x \, dx \). Integrate \( dv \): \( v = \frac{1}{2} e^{x^2} \).
04

Apply Integration by Parts

Substitute into the integration by parts formula: \[ \int x^3 e^{x^2} \, dx = \frac{1}{2} x^2 e^{x^2} - \int \frac{1}{2} e^{x^2} \cdot 2x \, dx \].
05

Simplify the Integral Expression

Simplify the remaining integral: \( \int e^{x^2} x \, dx \) is solved using substitution. Let \( t = x^2 \), then \( dt = 2x \, dx \), hence \( x \, dx = \frac{1}{2} \, dt \).
06

Substitute and Solve

Substitute \( x \, dx \) in the integral: \[ \int e^t \frac{1}{2} \, dt = \frac{1}{2} \int e^t \, dt = \frac{1}{2} e^t + C \]. Substitute back \( t = x^2 \).
07

Write Final Result

Combine and write the result as: \[ \frac{1}{2} x^2 e^{x^2} - \frac{1}{2} e^{x^2} + C \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Integration by Parts
Integration by parts is a powerful technique for evaluating integrals of products of functions. It is often used when the integral contains the product of a polynomial function and an exponential, trigonometric, or logarithmic function. The method is based on the product rule for differentiation and is articulated through the formula: \[ \int u \, dv = uv - \int v \, du \]Here are the important steps in using integration by parts:
  • Identify parts: Choose which part of the integrand will be \( u \) and which will be \( dv \). Commonly, \( u \) is chosen as the polynomial component because its derivative simplifies quickly, and \( dv \) is the remaining factor.
  • Differentiation and Integration: Differentiate \( u \) to find \( du \), and integrate \( dv \) to obtain \( v \).
  • Apply the Formula: Substitute \( u \), \( du \), \( v \), and \( dv \) into the integration by parts formula and solve for the integral.
Apply these steps to tackle complex integrals, like mixing polynomials and exponentials.
Substitution Method
The substitution method, also known as "u-substitution," is a simple yet essential integration technique. It is mainly used to simplify integrals by transforming them into an easier form by substituting part of the integral with a new variable. To effectively use substitution:
  • Identify the substitution: Look for a part of the integral whose derivative is present elsewhere in the integral. This often involves functions inside functions, like \( e^{x^2} \).
  • Make the substitution: Let \( t = g(x) \) where \( g(x) \) is the part of the function to replace. Differentiate it to find \( dt = g'(x) \, dx \).
  • Solve the new integral: Replace the identified parts in the integral with \( t \) and \( dt \). Solve the simplified integral with respect to \( t \).
  • Convert back: Once you have calculated the integral with respect to \( t \), substitute back the original variable \( x \) to obtain the final answer.
The substitution method works incredibly well for integrals involving exponential functions like \( e^{x^2} \), as shown in the previously solved integral.
Exponential Functions
Exponential functions, represented as \( e^{x} \) or variations like \( e^{x^2} \), occur frequently in calculus due to their unique properties. They are a specific type of function where the rate of growth of the function is proportional to its value, making them quite significant in mathematics. Here are key points about exponential functions:
  • The base \( e \) is an irrational number approximately equal to 2.71828, known as Euler's number.
  • Exponential functions exhibit continuous and persistent growth, applicable in fields like population dynamics, compound interest, and heat dissipation.
  • In integration, exponential functions are significant because they seamlessly integrate without change, which is why \( \int e^{x} \, dx = e^{x} + C \).
  • When exponentiated as \( e^{x^2} \), methods like substitution are used during integration to simplify and solve the integrals.
Understanding the characteristics of exponential functions is crucial in integration and calculus, given their wide applications and integral-friendly properties.

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