Integrate e^(-t/r) WITH RESPECT TO dt?
d^ [(-1/r) t] dt
-re^[(-1/r) t] + c
or,
(-1/r) e^[(-1/r) t]
since d/dt e^(kt) = ke^(kt)
To integrate the function e^(-t/r) with respect to dt, we can use the power rule of integration. However, before we do that, it's important to note that the function e^(-t/r) does not have any explicit bounds of integration mentioned. Therefore, we will assume that we are integrating over the entire range of t.
The power rule states that integrating a function of the form e^(kx) with respect to x gives (1/k) * e^(kx) + C, where C is the constant of integration.
In this case, we have e^(-t/r), so we can rewrite it as e^((1/r) * -t). Now we can use the power rule of integration:
∫ e^((1/r) * -t) dt = (-r) * e^((1/r) * -t) + C
Therefore, the integral of e^(-t/r) with respect to dt is (-r) * e^((1/r) * -t) + C.
Please note that if you have specific bounds of integration, you need to evaluate the integral at those bounds and subtract the value of the integral at the lower bound from the value of the integral at the upper bound.