Evaluate the definite integral: ${\int }_0^3(1+7x) dx$.

• A. $21$
• B. $24$
• C. $33$
• D. $30$

Answer: (C)

Verified step by step guidance

1. Step 1: Recall the formula for evaluating a definite integral: \( \int_a^b f(x) \, dx = F(b) - F(a) \), where \( F(x) \) is the antiderivative of \( f(x) \).
2. Step 2: Identify the function to integrate: \( f(x) = 1 + 7x \). Break it into two simpler terms: \( \int_0^3 1 \, dx + \int_0^3 7x \, dx \).
3. Step 3: Compute the antiderivative of each term. For \( \int 1 \, dx \), the antiderivative is \( x \). For \( \int 7x \, dx \), the antiderivative is \( \frac{7x^2}{2} \).
4. Step 4: Apply the limits of integration (from 0 to 3) to each antiderivative. Substitute \( x = 3 \) and \( x = 0 \) into \( x \) and \( \frac{7x^2}{2} \).
5. Step 5: Add the results from the two terms together to find the value of the definite integral. This will give the final result.