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

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

Evaluate

$$\frac{e\left(e^{5}-e^{2}+3e-3\right)}{3}\approx 132.451859794$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x^{2}+1\mathrm{d}x$$

Integrate the sum term by term.

$$\int x^{2}\mathrm{d}x+\int 1\mathrm{d}x$$

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}.

$$\frac{x^{3}}{3}+\int 1\mathrm{d}x$$

Find the integral of 1 using the table of common integrals rule \int a\mathrm{d}x=ax.

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

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.

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

Simplify.

$$\frac{e\left(e^{5}+3e-e^{2}-3\right)}{3}$$