Determine whether the integral is divergent or convergent. If it is convergent, evaluate it and enter that value as your answer. If it diverges to infinity, state your answer as "INF" (without the quotation marks). If it diverges to negative infinity, state your answer as "MINF". If it diverges without being infinity or negative infinity, state your answer as "DIV".
The integral from 0 to infinity of (3e^-x)dx
To determine whether the integral is convergent or divergent, we need to evaluate whether the integral from 0 to infinity of (3e^-x)dx converges.
To evaluate this integral, we can proceed as follows:
∫(3e^-x)dx = -3e^-x
Now we can evaluate the definite integral:
∫[0 to infinity](3e^-x)dx = lim[a->∞](-3e^-a) - (-3e^0)
Taking the limit as a approaches infinity, we have:
lim[a->∞](-3e^-a) = -3e^-∞ = -3(0) = 0
(-3e^0) = -3(1) = -3
Therefore, the value of the integral is 0 - (-3) = 3.
Hence, the integral converges and the value is 3.