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\int x^{4}-\frac{x^{4}}{2}\mathrm{d}x
Evaluate the indefinite integral first.
\int x^{4}\mathrm{d}x+\int -\frac{x^{4}}{2}\mathrm{d}x
Integrate the sum term by term.
\int x^{4}\mathrm{d}x-\frac{\int x^{4}\mathrm{d}x}{2}
Factor out the constant in each of the terms.
\frac{x^{5}}{5}-\frac{\int x^{4}\mathrm{d}x}{2}
Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{4}\mathrm{d}x with \frac{x^{5}}{5}.
\frac{x^{5}}{5}-\frac{x^{5}}{10}
Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{4}\mathrm{d}x with \frac{x^{5}}{5}. Multiply -\frac{1}{2} times \frac{x^{5}}{5}.
\frac{x^{5}}{10}
Simplify.
\frac{1^{5}}{10}-\frac{1}{10}\times \left(\frac{1}{2}\times 2^{\frac{1}{2}}\right)^{5}
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{1}{10}-\frac{\sqrt{2}}{80}
Simplify.