Solve ∫ (from n to 1) of 3x

$$\int _ { n } ^ { 1 } 3 x$$

Evaluate

$$\frac{3-3n^{2}}{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 3x\mathrm{d}x$$

Factor out the constant using \int af\left(x\right)\mathrm{d}x=a\int f\left(x\right)\mathrm{d}x.

$$3\int x\mathrm{d}x$$

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}.

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

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.

$$\frac{3}{2}\times 1^{2}-\frac{3}{2}n^{2}$$

Simplify.

$$\frac{-3n^{2}+3}{2}$$

Differentiate w.r.t. n

$$-3n$$