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

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. \[ \int_{0}^{1} \sqrt[3]{1+7 x} d x \]
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Solution

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Solution Steps

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 \(F(x)\) of \(f(x)\)

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.

Final Answer

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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