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\int _{1}^{4}\left(\sqrt{x}\right)^{2}-\sqrt{x}\mathrm{d}x
Use the distributive property to multiply \sqrt{x} by \sqrt{x}-1.
\int _{1}^{4}x-\sqrt{x}\mathrm{d}x
Calculate \sqrt{x} to the power of 2 and get x.
\int x-\sqrt{x}\mathrm{d}x
Evaluate the indefinite integral first.
\int x\mathrm{d}x+\int -\sqrt{x}\mathrm{d}x
Integrate the sum term by term.
\int x\mathrm{d}x-\int \sqrt{x}\mathrm{d}x
Factor out the constant in each of the terms.
\frac{x^{2}}{2}-\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\mathrm{d}x with \frac{x^{2}}{2}.
\frac{x^{2}}{2}-\frac{2x^{\frac{3}{2}}}{3}
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. Multiply -1 times \frac{2x^{\frac{3}{2}}}{3}.
\frac{4^{2}}{2}-\frac{2}{3}\times 4^{\frac{3}{2}}-\left(\frac{1^{2}}{2}-\frac{2}{3}\times 1^{\frac{3}{2}}\right)
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{17}{6}
Simplify.