Solve ∫ (from 0 to 1) of (x^2-e^x) wrt x

$$\int _ { 0 } ^ { 1 } ( x ^ { 2 } - e ^ { x } ) d x$$

Evaluate

$$\frac{4}{3}-e\approx -1.384948495$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{2}-e^{x}\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{2}\mathrm{d}x+\int -e^{x}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int x^{2}\mathrm{d}x-\int e^{x}\mathrm{d}x$$

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}.

$$\frac{x^{3}}{3}-\int e^{x}\mathrm{d}x$$

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

$$\frac{x^{3}}{3}-e^{x}$$

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.

$$\frac{1^{3}}{3}-e^{1}-\left(\frac{0^{3}}{3}-e^{0}\right)$$

Simplify.

$$\frac{4}{3}-e$$