Solve ∫ (from x to 3) of y^2+2 wrt y

$$\int _ { x } ^ { 3 } y ^ { 2 } + 2 d y$$

Evaluate

$$-\frac{x^{3}}{3}-2x+15$$

Solution Steps

Evaluate the indefinite integral first.

$$\int y^{2}+2\mathrm{d}y$$

Integrate the sum term by term.

$$\int y^{2}\mathrm{d}y+\int 2\mathrm{d}y$$

Since \int y^{k}\mathrm{d}y=\frac{y^{k+1}}{k+1} for k\neq -1, replace \int y^{2}\mathrm{d}y with \frac{y^{3}}{3}.

$$\frac{y^{3}}{3}+\int 2\mathrm{d}y$$

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

$$\frac{y^{3}}{3}+2y$$

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.

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

Simplify.

$$15-\frac{x^{3}}{3}-2x$$

Differentiate w.r.t. x

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