Find the Norm(Inner product)

Question

Find $\|xe^{-x}\|$(Find the Norm with respect to the inner product below).

The inner product is defined as $\int_{-1}^1 f(x)g(x)dx$.

Answer 1

Know that, Norm: <p , p> = $\sqrt{(p)^2}$ , where p = $xe^{-x}$

Step 1: First, Calculate the $\int^{1}_{-1} (xe^{-x})^2 dx$.

= $ \frac{e^2}4 - \frac{5e^{-2}}4$.

Step 2: Next, get the square root of $\frac{e^2}4 - \frac{5e^{-2}}4$.

Answer: $\frac{\sqrt{e^4 - 5}}{2e}$.