Problem 14 - Essential Calculus

Question

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{1}^{\infty} \frac{4}{\sqrt[4]{x}} d x $$

Step-by-Step Solution

Answer

The integral diverges.

Step 1: Rewrite the function

To work with the integral more easily, we can write the integrand in a form of negative power. We rewrite \( \frac{4}{\sqrt[4]{x}} \) as \( 4x^{-1/4} \). Thus, the integral becomes \( \int_{1}^{\infty} 4x^{-1/4} dx \).

Step 2: Use Power Rule for Integration

The power rule of integration states that the integral of \( x^n \) is \( \frac{x^{n+1}}{n+1} + C \) where C is the constant of integration. Applying this rule to our integral, we get \( \int 4x^{-1/4} dx = 4 \frac{x^{(-1/4 + 1)}}{-1/4 + 1} = 4x^{3/4} * 4/3 = \frac{16}{3}x^{3/4} \).

Step 3: Evaluate the improper integral

Now, we evaluate the improper integral from 1 to ∞. We do this by taking the limit as t approaches ∞ of the definite integral from 1 to t. This is represented as \( \lim_{t \to \infty} \int_{1}^{t} 4x^{-1/4} dx \). Substituting the antiderivative we have: \( \lim_{t \to \infty} [\frac{16}{3}t^{3/4} - \frac{16}{3}] \).

Step 4: Apply the limit

When we substitute ∞ into \( t^{3/4} \), the term becomes infinite, suggesting that the whole limit is infinite. Therefore, the integral does not converge. Instead, it diverges.