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$\integral{(2 \exponential{x}{3} - 4 x + 1)}{x} $
تقدير القيمة
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تفاضل w.r.t. x
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\int 2x^{3}\mathrm{d}x+\int -4x\mathrm{d}x+\int 1\mathrm{d}x
Integrate the sum term by term.
2\int x^{3}\mathrm{d}x-4\int x\mathrm{d}x+\int 1\mathrm{d}x
Factor out the constant in each of the terms.
\frac{x^{4}}{2}-4\int x\mathrm{d}x+\int 1\mathrm{d}x
Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}. اضرب 2 في \frac{x^{4}}{4}.
\frac{x^{4}}{2}-2x^{2}+\int 1\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}. اضرب -4 في \frac{x^{2}}{2}.
\frac{x^{4}}{2}-2x^{2}+x
Find the integral of 1 using the table of common integrals rule \int a\mathrm{d}x=ax.
\frac{x^{4}}{2}-2x^{2}+x+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.