In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.
∫₋∞⁰ x² e^(x³) dx

검증된 단계별 안내

Identify the integral as an improper integral because the lower limit is negative infinity: \(\int_{-\infty}^0 x^2 e^{x^3} \, dx\).

Consider the behavior of the integrand \(x^2 e^{x^3}\) as \(x \to -\infty\) to determine if the integral converges. Since \(x^3\) tends to \(-\infty\) as \(x \to -\infty\), analyze the exponential term \(e^{x^3}\) in this limit.

Use substitution to simplify the integral. Let \(t = x^3\), then compute \(dt\) in terms of \(dx\): \(dt = 3x^2 dx\), which implies \(x^2 dx = \frac{dt}{3}\).

Rewrite the integral in terms of \(t\) using the substitution: change the limits accordingly (when \(x = -\infty\), \(t = -\infty\); when \(x = 0\), \(t = 0\)), so the integral becomes \(\int_{-\infty}^0 e^t \frac{dt}{3}\).

Evaluate the integral \(\frac{1}{3} \int_{-\infty}^0 e^t dt\) by finding the antiderivative of \(e^t\) and then applying the limits to determine convergence and the value of the integral.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Improper Integrals

Improper integrals involve integration over infinite intervals or integrands with infinite discontinuities. To evaluate them, limits are used to define the integral as a limit of definite integrals over finite intervals. Determining convergence means checking if this limit exists and is finite.

Behavior of Exponential Functions with Polynomial Exponents

The function e^(x³) combines exponential growth or decay with a cubic polynomial in the exponent. For negative x, x³ is negative and large in magnitude, causing e^(x³) to approach zero rapidly, which affects the convergence of the integral when multiplied by x².

Techniques for Evaluating Improper Integrals

Evaluating improper integrals often requires substitution to simplify the integrand or integration by parts. Recognizing suitable substitutions, such as setting u = x³, can transform the integral into a more manageable form, enabling direct evaluation or application of known integral results.

In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.∫₋∞⁰ x² e^(x³) dx