Solve ∫ (from - 1 to 0) of -we^ktdw

$$\int _ { - 1 } ^ { 0 } - w e ^ { k t } d w$$

Evaluate

$$\frac{e^{kt}}{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int -we^{kt}\mathrm{d}w$$

Factor out the constant using \int af\left(w\right)\mathrm{d}w=a\int f\left(w\right)\mathrm{d}w.

$$-e^{kt}\int w\mathrm{d}w$$

Since \int w^{k}\mathrm{d}w=\frac{w^{k+1}}{k+1} for k\neq -1, replace \int w\mathrm{d}w with \frac{w^{2}}{2}.

$$-e^{kt}\times \frac{w^{2}}{2}$$

Simplify.

$$-\frac{e^{kt}w^{2}}{2}$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$-\frac{1}{2}e^{kt}\times 0^{2}+\frac{1}{2}e^{kt}\left(-1\right)^{2}$$

Simplify.

$$\frac{e^{kt}}{2}$$

Differentiate w.r.t. k

$$\frac{te^{kt}}{2}$$