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Differentiate w.r.t. x
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\int 5x^{4}\mathrm{d}x+\int -3x^{2}\mathrm{d}x+\int \frac{10}{x}\mathrm{d}x+\int \frac{7}{x^{2}}\mathrm{d}x
Integrate the sum term by term.
5\int x^{4}\mathrm{d}x-3\int x^{2}\mathrm{d}x+10\int \frac{1}{x}\mathrm{d}x+7\int \frac{1}{x^{2}}\mathrm{d}x
Factor out the constant in each of the terms.
x^{5}-3\int x^{2}\mathrm{d}x+10\int \frac{1}{x}\mathrm{d}x+7\int \frac{1}{x^{2}}\mathrm{d}x
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 5 times \frac{x^{5}}{5}.
x^{5}-x^{3}+10\int \frac{1}{x}\mathrm{d}x+7\int \frac{1}{x^{2}}\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 -3 times \frac{x^{3}}{3}.
x^{5}-x^{3}+10\ln(|x|)+7\int \frac{1}{x^{2}}\mathrm{d}x
Use \int \frac{1}{x}\mathrm{d}x=\ln(|x|) from the table of common integrals to obtain the result.
x^{5}-x^{3}+10\ln(|x|)-\frac{7}{x}
Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int \frac{1}{x^{2}}\mathrm{d}x with -\frac{1}{x}. Multiply 7 times -\frac{1}{x}.
x^{5}-x^{3}+10\ln(|x|)-\frac{7}{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.