Evaluate $\int x e^{2 x} d x$ using integration by parts.

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Solve the first-order differential equation $\frac{d y}{d x} = \frac{x}{y}$ with initial condition $y (0) = 2$ ( $y > 0$ ).
Evaluate the exact value of the integral $\int_{0}^{1} \frac{1}{x ^{2} + 1} d x$ .
A continuous random variable $X$ has probability density function $f (x) = \frac{3}{2} (1 - x^{2})$ for $0 \leq x \leq 1$ , and $0$ otherwise. Find…
The volume of a solid of revolution formed by rotating the region under $y = x e^{x}$ from $x = 0$ to $x = 1$ through $2 π$ radians ab…
The number of emails received by a support desk in an hour follows a Poisson distribution with mean $λ = 3$ . Find the probability that…