Solve ∫ (from 3 to 4) of x^2+y/2 wrt y

$$\int _ { 3 } ^ { 4 } x ^ { 2 } + \frac { y } { 2 } d y$$

Evaluate

$$x^{2}+\frac{7}{4}$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

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

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

$$x^{2}y+\frac{y^{2}}{4}$$

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.

$$x^{2}\times 4+\frac{4^{2}}{4}-\left(x^{2}\times 3+\frac{3^{2}}{4}\right)$$

Simplify.

$$x^{2}+\frac{7}{4}$$

Differentiate w.r.t. x

$$2x$$