Use the Evaluation Theorem to find the exact value of the integral
�ç 7 1 1/5x(dx)
To find the exact value of the integral ∫ [1, 7] (1/5x) dx using the Evaluation Theorem, also known as the Second Fundamental Theorem of Calculus, we need to follow these steps:
Step 1: Evaluate the Antiderivative
First, we need to find the antiderivative of the function (1/5x). The antiderivative of (1/5x) with respect to x is ln(|5x|), where ln represents the natural logarithm and | | denotes the absolute value.
Step 2: Evaluate F(x) at the upper bound
The Evaluation Theorem states that to evaluate the integral, we need to subtract the value of the antiderivative at the lower bound from the value of the antiderivative at the upper bound.
F(7) = ln(|5(7)|) = ln(35)
Step 3: Evaluate F(x) at the lower bound
F(1) = ln(|5(1)|) = ln(5)
Step 4: Evaluate the Integral
Finally, we subtract the value of the antiderivative at the lower bound from the value of the antiderivative at the upper bound.
∫ [1, 7] (1/5x) dx = F(7) - F(1) = ln(35) - ln(5)
To simplify further, we can use the logarithmic property: ln(a) - ln(b) = ln(a / b).
∫ [1, 7] (1/5x) dx = ln(35) - ln(5) = ln(35/5) = ln(7)
Thus, the exact value of the integral is ln(7).