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

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

Evaluate

$$\frac{50}{3}\approx 16.666666667$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

$$\frac{x^{3}}{6}+3\int x\mathrm{d}x$$

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

$$\frac{x^{3}}{6}+\frac{3x^{2}}{2}$$

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.

$$\frac{4^{3}}{6}+\frac{3}{2}\times 4^{2}-\left(\frac{3^{3}}{6}+\frac{3}{2}\times 3^{2}\right)$$

Simplify.

$$\frac{50}{3}$$