Solve ∫ (2+2t)dt

$$\int ( 2 + 2 t ) d t$$

Evaluate

$$t^{2}+2t+С$$

Solution Steps

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

Find the integral of 2 using the table of common integrals rule \int a\mathrm{d}t=at.

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

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}.

$$2t+t^{2}$$

If F\left(t\right) is an antiderivative of f\left(t\right), then the set of all antiderivatives of f\left(t\right) is given by F\left(t\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$2t+t^{2}+С$$

Differentiate w.r.t. t

$$2\left(t+1\right)$$