Solve ∫ (from 1 to 2) of (15x^2-12x) wrt x

$$\int_{ 1 }^{ 2 } (15 { x }^{ 2 } -12x) d x$$

Evaluate

$$17$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 15x^{2}-12x\mathrm{d}x$$

Integrate the sum term by term.

$$\int 15x^{2}\mathrm{d}x+\int -12x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$15\int x^{2}\mathrm{d}x-12\int x\mathrm{d}x$$

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}. Multiply 15 times \frac{x^{3}}{3}.

$$5x^{3}-12\int x\mathrm{d}x$$

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}. Multiply -12 times \frac{x^{2}}{2}.

$$5x^{3}-6x^{2}$$

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.

$$5\times 2^{3}-6\times 2^{2}-\left(5\times 1^{3}-6\times 1^{2}\right)$$

Simplify.

$$17$$