4. Evaluate ∫ (from 0 to 1) (1/x^(1/5)) dx after writing the integral as a limit.

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Rewrite the integral \( \int_0^1 \frac{1}{x^{1/5}} \, dx \) as \( \int_0^1 x^{-1/5} \, dx \) to express the integrand with a negative exponent, which is easier to integrate.

Recognize that the integrand \( x^{-1/5} \) is undefined at \( x = 0 \) because it involves division by zero, so the integral is an improper integral and must be expressed as a limit.

Express the integral as a limit approaching 0 from the right: \[ \lim_{t \to 0^+} \int_t^1 x^{-1/5} \, dx \]. This handles the improper behavior at the lower limit.

Find the antiderivative of \( x^{-1/5} \) using the power rule for integration: \[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \] where \( n = -\frac{1}{5} \). So, \[ \int x^{-1/5} \, dx = \frac{x^{4/5}}{4/5} + C = \frac{5}{4} x^{4/5} + C \].

Evaluate the definite integral by substituting the limits into the antiderivative and then take the limit as \( t \to 0^+ \): \[ \lim_{t \to 0^+} \left( \frac{5}{4} (1)^{4/5} - \frac{5}{4} t^{4/5} \right) \].

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주요 개념

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

Improper Integrals and Limits

An integral is improper if the integrand is unbounded or the interval is infinite. To evaluate such integrals, rewrite them as limits approaching the problematic point, ensuring the integral converges or diverges.

Power Rule for Integration

The power rule states that ∫ x^n dx = (x^(n+1)) / (n+1) + C for n ≠ -1. This rule helps integrate functions with variable exponents, such as x^(-1/5), by increasing the exponent by one and dividing by the new exponent.

Evaluating Definite Integrals

Definite integrals compute the net area under a curve between two limits. After finding the antiderivative, substitute the upper and lower limits and subtract to find the integral's value.