Solve ∫ (from 0 to x) of 2tdt

$$\int _ { 0 } ^ { x } 2 t d t$$

Evaluate

$$x^{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 2t\mathrm{d}t$$

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

$$2\int t\mathrm{d}t$$

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}. Multiply 2 times \frac{t^{2}}{2}.

$$t^{2}$$

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.

$$x^{2}-0^{2}$$

Simplify.

$$x^{2}$$

Differentiate w.r.t. x

$$2x$$