Solve ∫ (from 0 to 1) of (4x^2+sqrt{x}) wrt x

$$\int _ { 0 } ^ { 1 } ( 4 x ^ { 2 } + \sqrt { x } ) d x$$

Evaluate

$$2$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 4x^{2}+\sqrt{x}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 4x^{2}\mathrm{d}x+\int \sqrt{x}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$4\int x^{2}\mathrm{d}x+\int \sqrt{x}\mathrm{d}x$$

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 4 times \frac{x^{3}}{3}.

$$\frac{4x^{3}}{3}+\int \sqrt{x}\mathrm{d}x$$

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.

$$\frac{4x^{3}+2x^{\frac{3}{2}}}{3}$$

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.

$$\frac{4}{3}\times 1^{3}+\frac{2}{3}\times 1^{\frac{3}{2}}-\left(\frac{4}{3}\times 0^{3}+\frac{2}{3}\times 0^{\frac{3}{2}}\right)$$

Simplify.

$$2$$