Question

evaluate the integral.

\\( \int_{\ln 5}^{\ln 6} e^x \\, dx \\)

\\( \int_{\ln 5}^{\ln 6} e^x \\, dx = \square \\)

Explanation:

Step1: Recall the antiderivative of \( e^x \)

The antiderivative of \( e^x \) with respect to \( x \) is \( e^x \) itself, since the derivative of \( e^x \) is \( e^x \). So, by the Fundamental Theorem of Calculus, the definite integral \( \int_{a}^{b} e^x dx = e^x \big|_{a}^{b} \).

Step2: Apply the Fundamental Theorem of Calculus

We need to evaluate \( e^x \) at the upper limit \( \ln 6 \) and the lower limit \( \ln 5 \), then subtract. So, \( e^x \big|_{\ln 5}^{\ln 6} = e^{\ln 6} - e^{\ln 5} \).

Step3: Simplify using the property of exponents and logarithms

Recall that \( e^{\ln a} = a \) for any positive real number \( a \). So, \( e^{\ln 6} = 6 \) and \( e^{\ln 5} = 5 \). Then, \( 6 - 5 = 1 \).

Answer:

\( 1 \)