Solve ∫ (from 1 to 100) of x^2+n

$$\int _ { 1 } ^ { 100 } x ^ { 2 } + n$$

Evaluate

$$99\left(n+3367\right)$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

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}.

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

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

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

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.

$$\frac{100^{3}}{3}+n\times 100-\left(\frac{1^{3}}{3}+n\times 1\right)$$

Simplify.

$$333333+99n$$

Differentiate w.r.t. n

$$99$$