Solve ∫ (from 2 to 3) of x(x+5x)

$$\int _ { 2 } ^ { 3 } x ( x + 5 x )$$

Evaluate

$$38$$

Solution Steps

Combine x and 5x to get 6x.

$$\int _{2}^{3}x\times 6x\mathrm{d}x$$

Multiply x and x to get x^{2}.

$$\int _{2}^{3}x^{2}\times 6\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 6x^{2}\mathrm{d}x$$

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

$$6\int x^{2}\mathrm{d}x$$

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}. Multiply 6 times \frac{x^{3}}{3}.

$$2x^{3}$$

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.

$$2\times 3^{3}-2\times 2^{3}$$

Simplify.

$$38$$