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Chapter 5: Problem 54

Evaluate. $$ \int_{0}^{1} 3 x^{2} e^{x^{3}} d x $$

Short Answer

Expert verified
The value of the integral is \(e - 1\).

Step by step solution

01

Recognize the Integration Technique

Notice that the integral involves a function of the form \(x^n e^{g(x)}\), where \(g(x)\) is a polynomial. This is a perfect candidate for substitution.
02

Choose a Substitution

Let \(u = x^3\). Then, calculate the differential \(du\), where \(du = 3x^2 dx\).
03

Rewrite the Integral in Terms of u

Substitute \(u = x^3\) and \(du = 3x^2 dx\) into the integral, which changes the limits as well: when \(x = 0\), \(u = 0\); when \(x = 1\), \(u = 1\). The integral becomes: \[ \int_{0}^{1} e^u du. \]
04

Integrate with Respect to u

Evaluate the integral \(\int_{0}^{1} e^u du\), which gives \( [e^u]_{0}^{1}. \)
05

Evaluate the Definite Integral

Calculate \( [e^u]_{0}^{1} = e^1 - e^0 = e - 1. \)

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Key Concepts

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

Integration Techniques
Integration is a fundamental tool in calculus, used to find quantities like area under a curve. Recognizing which technique to use is crucial. For example, when you see an integral with functions multiplying or nested, such as \( \int_{0}^{1} x^n e^{g(x)} dx \), substitution might be just the right approach. Other techniques include \(integration by parts\), \(partial fractions\), and sometimes even \(trigonometric identities\). Each method simplifies complex integrals into more accessible, solvable forms.
U-Substitution
In solving integrals, one powerful method is \(u\)-substitution. This technique involves substituting part of the integral with a new variable \(u\) to simplify it. For instance, in the integral \( \int_{0}^{1} 3 x^{2} e^{x^{3}} dx \), you select \( u = x^3 \) because its derivative \( du = 3x^2 dx \) appears in the integrand. The goal is to transform the integral into terms of \(u\), making it easier to evaluate. After substitution, do not forget to change the limits of integration if it's a definite integral.
Evaluating Integrals
Once the integral is rewritten in terms of \(u\), the next step is evaluation. Take \( \int_{0}^{1} e^u du \), where continuous and differentiable functions of \(u\) are straightforward to integrate. Evaluate it over the transformation limits left from the substitution. For \( e^u \), integrate and apply the limits: \( [e^u]_{0}^{1} \). Ultimately, you're left with the result \( e - 1 \), giving you the integral's value. By breaking down complex integrations into simpler steps, you can solve intricate problems seamlessly.

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Most popular questions from this chapter

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