Solve ∫ (from 0 to 1) of e^tx

$$\int _ { 0 } ^ { 1 } e ^ { t } x$$

Evaluate

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

Solution Steps

Evaluate the indefinite integral first.

$$\int e^{t}x\mathrm{d}x$$

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

$$e^{t}\int x\mathrm{d}x$$

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

$$e^{t}\times \frac{x^{2}}{2}$$

Simplify.

$$\frac{e^{t}x^{2}}{2}$$

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.

$$\frac{1}{2}e^{t}\times 1^{2}-\frac{1}{2}e^{t}\times 0^{2}$$

Simplify.

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

Differentiate w.r.t. t

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