Solve ∫ (from 0 to t) of exp(ax) wrt x

$$\int _ { 0 } ^ { t } \exp ( a x ) d x$$

Evaluate

$$\frac{eapt^{3}}{3}$$

Solution Steps

Multiply x and x to get x^{2}.

$$\int _{0}^{t}ex^{2}pa\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int ex^{2}pa\mathrm{d}x$$

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

$$epa\int x^{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}.

$$epa\times \frac{x^{3}}{3}$$

Simplify.

$$\frac{epax^{3}}{3}$$

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.

$$\frac{1}{3}epat^{3}-\frac{1}{3}epa\times 0^{3}$$

Simplify.

$$\frac{epat^{3}}{3}$$

Differentiate w.r.t. p

$$\frac{eat^{3}}{3}$$