Solve f ( x ) = ∫ (from 1 to x ^ 2) of xe^tdt

$$f ( x ) = \int _ { 1 } ^ { x ^ { 2 } } x e ^ { t } d t$$

Evaluate

$$x\left(e^{x^{2}}-e\right)$$

Solution Steps

Evaluate the indefinite integral first.

$$\int xe^{t}\mathrm{d}t$$

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

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

Use \int e^{t}\mathrm{d}t=e^{t} from the table of common integrals to obtain the result.

$$xe^{t}$$

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.

$$xe^{x^{2}}-xe^{1}$$

Simplify.

$$x\left(e^{x^{2}}-e\right)$$

Differentiate w.r.t. x

$$2x^{2}e^{x^{2}}+e^{x^{2}}-e$$