Solve ∫ (from 1 to x) of 3tdt

$$\int _ { 1 } ^ { x } 3 t d t$$

Evaluate

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

Solution Steps

Evaluate the indefinite integral first.

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

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

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

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

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

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.

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

Simplify.

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

Differentiate w.r.t. x

$$3x$$