What is the value of A if $\int_{0}^{\ln 6} \frac{x}{e^{x}} d x=-\frac{\ln 6}{6}+\frac{A}{6}$ ?
A)5
B)6
C)7
D)None of the above.

Question

Quizplus · Accepted Answer

To solve the integral $\int_{0}^{\ln 6} \frac{x}{e^{x}} d x$, we can use integration by parts, where $u = x$ and $dv = e^{-x}dx$. This gives us $du = dx$ and $v = -e^{-x}$. Applying the integration by parts formula, we get:$$\int u dv = uv - \int v du = -xe^{-x} - \int -e^{-x} dx = -xe^{-x} + e^{-x} + C$$Evaluating this from $0$ to $\ln 6$, we get:$$[-xe^{-x} + e^{-x}]_{0}^{\ln 6} = [-(\ln 6)e^{-\ln 6} + e^{-\ln 6}] - [0 + 1] = -\frac{\ln 6}{6} + \frac{1}{6} - 1$$Given that $\int_{0}^{\ln 6} \frac{x}{e^{x}} d x=-\frac{\ln 6}{6}+\frac{A}{6}$, comparing the two expressions, we find that $A = 5$, since the $-1$ when added to the fraction $\frac{1}{6}$ gives us the $\frac{A}{6}$ term, and $A$ must be $5$ to satisfy the equation.

What Is the Value of a If ∫0ln⁡6xexdx=−ln⁡66+A6\int_{0}^{\ln 6} \frac{x}{e^{x}} d x=-\frac{\ln 6}{6}+\frac{A}{6}∫0ln6​exx​dx=−6ln6​+6A​

What Is the Value of a If $\int_{0}^{\ln 6} \frac{x}{e^{x}} d x=-\frac{\ln 6}{6}+\frac{A}{6}$