3.6 Q-23

Question Statement

Evaluate the following integral:

$$\int_{\frac{1}{8}}^{1} \frac{\left(x^{\frac{1}{3}} + 2\right)^2}{x^{\frac{2}{3}}} , dx$$

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Background and Explanation

This integral involves a rational expression with powers of $x$ and requires a substitution to simplify the integration process. To solve this, we need to:

• Expand the integrand: Simplify the expression inside the integral.
• Apply substitution: Use the cube root property and simplify the resulting terms.
• Integrate: Follow basic power rule integration techniques.

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Solution

Step 1: Expand the numerator

We begin by expanding the expression in the numerator, $(x^{\frac{1}{3}} + 2)^2$:

$$(x^{\frac{1}{3}} + 2)^2 = x^{\frac{2}{3}} + 4x^{\frac{1}{3}} + 4$$

So the integral becomes:

$$\int_{\frac{1}{8}}^{1} \frac{x^{\frac{2}{3}} + 4x^{\frac{1}{3}} + 4}{x^{\frac{2}{3}}} , dx$$

Step 2: Simplify the integrand

Now, divide each term in the numerator by $x^{\frac{2}{3}}$:

$$\frac{x^{\frac{2}{3}}}{x^{\frac{2}{3}}} + \frac{4x^{\frac{1}{3}}}{x^{\frac{2}{3}}} + \frac{4}{x^{\frac{2}{3}}}$$

This simplifies to:

$$1 + 4x^{-\frac{1}{3}} + 4x^{-\frac{2}{3}}$$

Thus, the integral becomes:

$$\int_{\frac{1}{8}}^{1} \left(1 + 4x^{-\frac{1}{3}} + 4x^{-\frac{2}{3}}\right) , dx$$

Step 3: Integrate each term

We can now integrate each term separately:

• First term: $\int 1 , dx = x$
• Second term: $\int 4x^{-\frac{1}{3}} , dx = 4 \times \frac{x^{\frac{2}{3}}}{\frac{2}{3}} = 6x^{\frac{2}{3}}$
• Third term: $\int 4x^{-\frac{2}{3}} , dx = 4 \times \frac{x^{\frac{1}{3}}}{\frac{1}{3}} = 12x^{\frac{1}{3}}$

Thus, the integral becomes:

$$\left[ x + 6x^{\frac{2}{3}} + 12x^{\frac{1}{3}} \right]_{\frac{1}{8}}^{1}$$

Step 4: Evaluate the definite integral

Now, we evaluate the expression at the limits $x = 1$ and $x = \frac{1}{8}$.

• When $x = 1$:

$$1 + 6(1)^{\frac{2}{3}} + 12(1)^{\frac{1}{3}} = 1 + 6 + 12 = 19$$

• When $x = \frac{1}{8}$: First, calculate the cube root:

$$\left(\frac{1}{8}\right)^{\frac{1}{3}} = \frac{1}{2}$$

Now, substitute into the expression:

$$\frac{1}{8} + 6 \times \left(\frac{1}{8}\right)^{\frac{2}{3}} + 12 \times \frac{1}{2}$$

We need $\left(\frac{1}{8}\right)^{\frac{2}{3}}$, which is $\frac{1}{4}$. So:

$$\frac{1}{8} + 6 \times \frac{1}{4} + 12 \times \frac{1}{2} = \frac{1}{8} + \frac{3}{2} + 6 = \frac{1}{8} + \frac{12}{8} + \frac{48}{8} = \frac{61}{8}$$

Step 5: Final result

Subtract the two results:

$$19 - \frac{61}{8} = \frac{152}{8} - \frac{61}{8} = \frac{91}{8}$$

Thus, the value of the integral is:

$$\boxed{\frac{91}{8}}$$

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Key Formulas or Methods Used

• Power Rule for Integration: $\int x^n , dx = \frac{x^{n+1}}{n+1}$ for $n \neq -1$.
• Definite Integral: The integral is evaluated over specific limits.

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Summary of Steps

1. Expand the numerator $(x^{\frac{1}{3}} + 2)^2$.
2. Simplify the integrand by dividing each term by $x^{\frac{2}{3}}$.
3. Integrate each term separately.
4. Evaluate the definite integral at the limits $x = 1$ and $x = \frac{1}{8}$.
5. Subtract the results to find the final answer: $\frac{91}{8}$.

The final answer is:

$$\boxed{\frac{91}{8}}$$