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

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

Evaluate

$$e^{2}+\frac{1}{3}\approx 7.722389432$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

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}.

$$\frac{x^{3}}{3}+\int e^{2}\mathrm{d}x$$

Find the integral of e^{2} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x^{3}}{3}+e^{2}x$$

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.

$$\frac{1^{3}}{3}+e^{2}\times 1-\left(\frac{0^{3}}{3}+e^{2}\times 0\right)$$

Simplify.

$$\frac{1}{3}+e^{2}$$