Problem 35 $$ \text { Use integration by ... [FREE SOLUTION]

Chapter 6: Problem 35

$$ \text { Use integration by parts to find each integral. } $$ $$ \begin{array}{l} \int x^{3} e^{x^{2}} d x \quad\left[\text { Hint }: \text { Take } u=x^{2}, \quad d v=x e^{x^{2}},\right.\\\ \text { and use a substitution to find } v \text { from } d v \text { . } \end{array} $$

Short Answer

Expert verified

The integral is $ \frac{1}{2} e^{x^2} (x^2 - 1) + C $.

Step by step solution

Choose Functions for Integration by Parts

For integration by parts, choose $ u = x^2 $ and $ dv = x e^{x^2} \, dx $. This requires that we find $ du $ and $ v $ before applying the integration by parts formula.

Differentiate and Integrate Separately

Differentiate $ u $ to get $ du = 2x \, dx $. Next, integrate $ dv = x e^{x^2} \, dx $. Let $ w = x^2 $, then $ dw = 2x \, dx $. Thus, the integral becomes $ \int e^w \frac{dw}{2} = \frac{1}{2} e^w + C = \frac{1}{2} e^{x^2} + C $, so $ v = \frac{1}{2} e^{x^2} $.

Apply Integration by Parts Formula

The integration by parts formula is $ \int u \, dv = uv - \int v \, du $. Substituting the parts, we get $ \int x^3 e^{x^2} \, dx = x^2 \cdot \frac{1}{2} e^{x^2} - \int \frac{1}{2} e^{x^2} \cdot 2x \, dx $. Simplifying, $ \int x^3 e^{x^2} \, dx = \frac{x^2}{2} e^{x^2} - \int x e^{x^2} \, dx $.

Resolve the Remaining Integral

The remaining integral $ \int x e^{x^2} \, dx $ is straightforward because it matches the integrated form from Step 2. Substitute $ w = x^2 $, then $ dw = 2x \, dx $, and the integral becomes $ \frac{1}{2} \int e^w \frac{dw}{2} = \frac{1}{2} e^w + C = \frac{1}{2} e^{x^2} + C $.

Combine Results

Now, combine the parts: $ \frac{x^2}{2} e^{x^2} - (\frac{1}{2} e^{x^2} + C) = \frac{x^2}{2} e^{x^2} - \frac{1}{2} e^{x^2} - C $. This simplifies to $ \frac{1}{2} e^{x^2} (x^2 - 1) - C $.

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Key Concepts

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

Understanding Integrals

Integrals are a fundamental concept in calculus dealing with the accumulation of quantities. Essentially, an integral calculates the area under a curve, defined by a function, across a specific interval. There are two main types of integrals: definite and indefinite.

**Indefinite Integrals:** These refer to integrals without specific limits of integration. The result is a general form plus a constant, represented mathematically as $ \int f(x) \, dx = F(x) + C $. The constant $ C $ is crucial because integration is the reverse process of differentiation, and for any function $ F(x) $, there can be many antiderivatives differing by a constant.

**Definite Integrals:** These involve specific upper and lower limits and represent the total area under the curve between those limits. It's written as $ \int_{a}^{b} f(x) \, dx $, where $ a $ and $ b $ are the limits. The result of a definite integral is a specific number, not a function.

Substitution Method in Integration

The substitution method is a powerful technique for simplifying the integration process, especially when faced with complex functions. This technique is akin to reversing the chain rule in differentiation. Here's how it works:
Choose a new variable, typically denoted by $ u $, and express part of the original integral in terms of $ u $. The goal is to make the integral easier to handle.

Determine $ du $ by differentiating $ u $ with respect to $ x $. This helps in replacing $ dx $ in the integral.

Convert the entire integral from terms of $ x $ to terms of $ u $. At this stage, the integral should be simpler, allowing for straightforward integration.

After integrating, revert back to the original variable $ x $.
In the original exercise, integration by parts was initially chosen, then substitution was used to handle the integral $ \int x e^{x^2} \, dx $. Substitution often complements other integration techniques when direct integration is challenging.

Definite and Indefinite Integrals

Definite and indefinite integrals, as mentioned, are two sides of the calculus coin. Each has its unique characteristics and applications.
**Indefinite Integrals** involve finding a general formula for the antiderivative of a function. They're specified without limits and always include a constant $ + C $. The integration by parts and substitution techniques often aim to solve indefinite integrals.

**Definite Integrals** provide the evaluated result of an area between the curve and the x-axis from one point to another. The evaluation of a definite integral uses the Fundamental Theorem of Calculus, simplifying integration by using antiderivatives.
In practical applications, definite integrals can determine things like areas, volumes, and central points, while indefinite integrals guide finding the antiderivatives necessary to solve those real-life problems.

91影视

$$ \text { Use integration by parts to find each integral. } $$ $$ \begin{array}{l} \int x^{3} e^{x^{2}} d x \quad\left[\text { Hint }: \text { Take } u=x^{2}, \quad d v=x e^{x^{2}},\right.\\\ \text { and use a substitution to find } v \text { from } d v \text { . } \end{array} $$

Short Answer

Step by step solution

Choose Functions for Integration by Parts

Differentiate and Integrate Separately

Apply Integration by Parts Formula

Resolve the Remaining Integral

Combine Results

Key Concepts

Understanding Integrals

Substitution Method in Integration

Definite and Indefinite Integrals

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