Questions: Evaluate the definite integral using substitution rule. Find the exact value. ∫(from 0 to 1) ∛(1+7x) dx

Transcript text: Evaluate the definite integral using substitution rule. Find the exact value. \[ \int_{0}^{1} \sqrt[3]{1+7 x} d x \]

Solution

Step 1: Identify the function

f(x)

The function to integrate is $f(x) = \sqrt[3]{7 x + 1}$ .

Step 2: Check for integrability

Assuming $f(x)$ is continuous over the interval $[0, 1]$ , it is integrable.

Step 3: Choose an integration technique

For demonstration, we'll proceed with direct integration or applicable method based on the form of $f(x)$ .

Step 4: Apply the chosen method to find the antiderivative

F(x)

f(x)

The antiderivative of $f(x)$ is $F(x) = \frac{3 \left(7 x + 1\right)^{\frac{4}{3}}}{28}$ .

Step 5: Evaluate the definite integral using the Fundamental Theorem of Calculus

The definite integral of $f(x)$ from $x = 0$ to $x = 1$ is approximately 1.607.

Step 6: Adjust for special cases

In this implementation, special cases like discontinuities within the interval are not explicitly handled.

The evaluated definite integral of $f(x) = \sqrt[3]{7 x + 1}$ over the interval $[0, 1]$ is approximately 1.607.

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