Solve ∫ (from 0 to 1) of intxe^x wrt xdy=

$$\int _ { 0 } ^ { 1 } \int x e ^ { x } d x d y =$$

Evaluate

$$xe^{x}-e^{x}+С$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \int xe^{x}\mathrm{d}x\mathrm{d}y$$

Find the integral of \int xe^{x}\mathrm{d}x using the table of common integrals rule \int a\mathrm{d}y=ay.

$$\int xe^{x}\mathrm{d}xy$$

Simplify.

$$\left(e^{x}x-e^{x}+С\right)y$$

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.

$$\left(e^{x}x-e^{x}+С\right)\times 1-\left(e^{x}x-e^{x}+С\right)\times 0$$

Simplify.

$$e^{x}x-e^{x}+С$$

Differentiate w.r.t. x

$$xe^{x}$$