Solve ∫ (from 0 to y) of (1+t)dt

$$\int _ { 0 } ^ { y } ( 1 + t ) d t$$

Evaluate

$$\frac{y\left(y+2\right)}{2}$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 1+t\mathrm{d}t$$

Integrate the sum term by term.

$$\int 1\mathrm{d}t+\int t\mathrm{d}t$$

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

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

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}.

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

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.

$$y+\frac{y^{2}}{2}-\left(0+\frac{0^{2}}{2}\right)$$

Simplify.

$$\frac{y\left(2+y\right)}{2}$$

Differentiate w.r.t. y

$$y+1$$