Evaluate the integrals. (a) integral_{-infinity}^{infinity} x^2 e^{-x^3} dx. (b) integral...


Evaluate the integrals.

(a) {eq}\displaystyle \int_{-\infty}^{\infty} x^2 e^{-x^3}\ dx {/eq}.

(b) {eq}\displaystyle \int _{-\infty}^1 e^{2 x}\ dx {/eq}.

(c) {eq}\displaystyle \int_{-\infty}^\infty e^{|x|}\ dx {/eq}.

Definite Integrals:

We will solve the problem where we will use the substitution method for problem b) and for part c) we will use the property of modulus function and then integrate and plug-in the bounds.

Answer and Explanation:

a) To solve the problem we will integrate:

{eq}\int_{-\infty}^{\infty}x^{2}e^{-x^{3}}dx\\ x^{3}=t\\ 3x^{2}dx=dt\\ =\frac{1}{3}\int_{-\infty}^{\infty}e^{-t}dt\\ =\infty {/eq}

b) Now let us solve:

{eq}\int_{-\infty}^{1}e^{2x}dx\\ =\frac{1}{2}e^{2} {/eq}

c) Now let u solve:

{eq}=\int_{-\infty}^{\infty}e^{|x|}dx\\ =\int_{-\infty}^{0}e^{-x}dx+\int_{0}^{\infty}e^{x}dx\\ =-e^{-x}+e^{x}\\ =\infty {/eq}

Learn more about this topic:

Evaluating Definite Integrals Using the Fundamental Theorem

from AP Calculus AB: Exam Prep

Chapter 16 / Lesson 2

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