Determine whether the integral is convergent or divergent. $\int_{0}^{33} 5 (x - 1)^{-1/5} \, dx$

If it is convergent, evaluate it. (If the quantity diverges, enter DIVERGES.)

Neetesh Kumar | December 10, 2024

Calculus Homework Help

This is the solution to Math 132
Assignment: 7.8 Question Number 4
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Step-by-step solution:

Step 1: Analyze the integral for convergence

The given integral is:

$\int_{0}^{33} 5 (x - 1)^{-1/5} \, dx$

The integrand has a potential issue at $x = 1$ because $(x - 1)^{-1/5}$ becomes undefined at this point.

Thus, we need to evaluate the behavior of the integral near $x = 1$ to determine if it is convergent or divergent.

Step 2: Break the integral into two parts

To handle the problematic point at $x = 1$, break the integral into two parts:

$\int_{0}^{33} 5 (x - 1)^{-1/5} \, dx = \int_{0}^{1} 5 (x - 1)^{-1/5} \, dx + \int_{1}^{33} 5 (x - 1)^{-1/5} \, dx.$

• Part 1: Analyze the integral $\int_{0}^{1} 5 (x - 1)^{-1/5} \, dx$.
• Part 2: Analyze the integral $\int_{1}^{33} 5 (x - 1)^{-1/5} \, dx$.

Step 3: Focus on the first part (near $x = 1$)

The integral $\int_{0}^{1} 5 (x - 1)^{-1/5} \, dx$ requires a substitution to evaluate its convergence.

Let $u = x - 1$, so $du = dx$. When $x = 0$, $u = -1$, and when $x = 1$, $u = 0$.

The integral becomes:

$\int_{-1}^{0} 5 u^{-1/5} \, du.$

The power of $u$, $-1/5$, is greater than $-1$.

Therefore, the integral converges near $u = 0$.

Step 4: Analyze the second part (from $x = 1$ to $x = 33$)

The integral $\int_{1}^{33} 5 (x - 1)^{-1/5} \, dx$ is well-defined because $(x - 1)^{-1/5}$ does not become undefined within the interval $[1, 33]$.

Thus, this part of the integral converges.

Step 5: Combine the results

Both parts of the integral converge, so the entire integral is convergent.

Step 6: Evaluate the integral

The original integral can now be evaluated: $\int_{0}^{33} 5 (x - 1)^{-1/5} \, dx.$

1. Solve $\int (x - 1)^{-1/5} \, dx$:

   The integral of $(x - 1)^{-1/5}$ is:

   $\frac{(x - 1)^{4/5}}{4/5} = \frac{5}{4} (x - 1)^{4/5}.$

2. Apply the limits $0$ to $33$:

   $5 \cdot \frac{5}{4} \left[ (33 - 1)^{4/5} - (0 - 1)^{4/5} \right].$

   Simplify:

   $\frac{25}{4} \left[ 32^{4/5} - (-1)^{4/5} \right].$

   Note that $(-1)^{4/5} = 1$ (since the fourth root is even).

   $\frac{25}{4} \left[ 32^{4/5} - 1 \right].$

Final Answer:

The integral is $\boxed{\text{convergent}}$, and its value is: $\boxed{\frac{25}{4} \left[ 32^{4/5} - 1 \right] \space \text{or} \space 93.75}$

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