Kerala KEAM 5 June 2024 Solved Paper

We are given the integral: $$\int x e^{-x} \, dx = M e^{-x} + C$$We are tasked with finding the value of $M$.Step 1: Use integration by parts to solve the integral. Recall the formula for integration by parts:$$\int u \, dv = uv - \int v \, du$$Let $u = x$ and $dv = e^{-x} \, dx$. Then, $du = dx$ and $v = -e^{-x}$.Step 2: Apply the integration by parts formula:$$\int x e^{-x} \, dx = -x e^{-x} - \int -e^{-x} \, dx$$Simplify: $$\int x e^{-x} \, dx = -x e^{-x} + \int e^{-x} \, dx$$The integral of $e^{-x}$ is $-e^{-x}$, so: $$\int x e^{-x} \, dx = -x e^{-x} - e^{-x} + C$$Step 3: Factor out $e^{-x}$:$$\int x e^{-x} \, dx = -(x + 1) e^{-x} + C$$Thus, comparing with the given equation $\int x e^{-x} \, dx = M e^{-x} + C$, we see that:$$M = -(x + 1)$$Thus, the correct answer is option (A).Quick Tip: Use integration by parts for integrals involving a product of functions. Choose $u$ as the polynomial term and $dv$ as the exponential term to simplify the integration.