Solve ∫ (from 1 to 4) of (sqrt{x}+2x) wrt x

$$\int _ { 1 } ^ { 4 } ( \sqrt { x } + 2 x ) d x$$

Evaluate

$$\frac{59}{3}\approx 19.666666667$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \sqrt{x}+2x\mathrm{d}x$$

Integrate the sum term by term.

$$\int \sqrt{x}\mathrm{d}x+\int 2x\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int \sqrt{x}\mathrm{d}x+2\int x\mathrm{d}x$$

Rewrite \sqrt{x} as x^{\frac{1}{2}}. Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{\frac{1}{2}}\mathrm{d}x with \frac{x^{\frac{3}{2}}}{\frac{3}{2}}. Simplify.

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

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 2 times \frac{x^{2}}{2}.

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

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.

$$\frac{2}{3}\times 4^{\frac{3}{2}}+4^{2}-\left(\frac{2}{3}\times 1^{\frac{3}{2}}+1^{2}\right)$$

Simplify.

$$\frac{59}{3}$$