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Evaluate
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Differentiate w.r.t. x
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\int 2\mathrm{d}x+\int 6x^{2}\mathrm{d}x
Integrate the sum term by term.
\int 2\mathrm{d}x+6\int x^{2}\mathrm{d}x
Factor out the constant in each of the terms.
2x+6\int x^{2}\mathrm{d}x
Find the integral of 2 using the table of common integrals rule \int a\mathrm{d}x=ax.
2x+2x^{3}
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 6 times \frac{x^{3}}{3}.
2x+2x^{3}+С
If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.