Solve ∫ (from 2 to x) of (3t^2-1)dt

$$\int _ { 2 } ^ { x } ( 3 t ^ { 2 } - 1 ) d t$$

Evaluate

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

Solution Steps

Evaluate the indefinite integral first.

$$\int 3t^{2}-1\mathrm{d}t$$

Integrate the sum term by term.

$$\int 3t^{2}\mathrm{d}t+\int -1\mathrm{d}t$$

Factor out the constant in each of the terms.

$$3\int t^{2}\mathrm{d}t+\int -1\mathrm{d}t$$

Since \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} for k\neq -1, replace \int t^{2}\mathrm{d}t with \frac{t^{3}}{3}. Multiply 3 times \frac{t^{3}}{3}.

$$t^{3}+\int -1\mathrm{d}t$$

Find the integral of -1 using the table of common integrals rule \int a\mathrm{d}t=at.

$$t^{3}-t$$

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.

$$x^{3}-x-\left(2^{3}-2\right)$$

Simplify.

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

Differentiate w.r.t. x

$$3x^{2}-1$$