Solve ∫ _{text{Scanneld}}^3(x^2+x) wrt x

$$\int _ { \text { Scanneld } } ^ { 3 } ( x ^ { 2 } + x ) d x$$

Evaluate

$$\frac{-n^{4}\left(2eSacdln^{2}+3\right)\left(eSacdl\right)^{2}+81}{6}$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

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}.

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

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}.

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

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.

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

Simplify.

$$\frac{27}{2}-\frac{a^{2}e^{2}\left(2Scean^{2}ld+3\right)S^{2}c^{2}n^{4}l^{2}d^{2}}{6}$$

Differentiate w.r.t. S

$$Sn^{4}\left(-eSacdln^{2}-1\right)\left(eacdl\right)^{2}$$