Solve ∫ (12e^t+7t)dt

$$\int ( 12 e ^ { t } + 7 t ) d t$$

Evaluate

$$12e^{t}+\frac{7t^{2}}{2}+С$$

Solution Steps

Integrate the sum term by term.

$$\int 12e^{t}\mathrm{d}t+\int 7t\mathrm{d}t$$

Factor out the constant in each of the terms.

$$12\int e^{t}\mathrm{d}t+7\int t\mathrm{d}t$$

Use \int e^{t}\mathrm{d}t=e^{t} from the table of common integrals to obtain the result.

$$12e^{t}+7\int t\mathrm{d}t$$

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

$$12e^{t}+\frac{7t^{2}}{2}$$

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.

$$12e^{t}+\frac{7t^{2}}{2}+С$$

Differentiate w.r.t. t

$$7t+12e^{t}$$