Solve ∫ (5x^4-3x^2+10/x+7/x^2) wrt x

$$\int ( 5 x ^ { 4 } - 3 x ^ { 2 } + \frac { 10 } { x } + \frac { 7 } { x ^ { 2 } } ) d x$$

Evaluate

$$10\ln(|x|)+x^{5}-x^{3}-\frac{7}{x}+С$$

Solution Steps

Integrate the sum term by term.

$$\int 5x^{4}\mathrm{d}x+\int -3x^{2}\mathrm{d}x+\int \frac{10}{x}\mathrm{d}x+\int \frac{7}{x^{2}}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$5\int x^{4}\mathrm{d}x-3\int x^{2}\mathrm{d}x+10\int \frac{1}{x}\mathrm{d}x+7\int \frac{1}{x^{2}}\mathrm{d}x$$

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}.

$$x^{5}-3\int x^{2}\mathrm{d}x+10\int \frac{1}{x}\mathrm{d}x+7\int \frac{1}{x^{2}}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply -3 times \frac{x^{3}}{3}.

$$x^{5}-x^{3}+10\int \frac{1}{x}\mathrm{d}x+7\int \frac{1}{x^{2}}\mathrm{d}x$$

Use \int \frac{1}{x}\mathrm{d}x=\ln(|x|) from the table of common integrals to obtain the result.

$$x^{5}-x^{3}+10\ln(|x|)+7\int \frac{1}{x^{2}}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int \frac{1}{x^{2}}\mathrm{d}x with -\frac{1}{x}. Multiply 7 times -\frac{1}{x}.

$$x^{5}-x^{3}+10\ln(|x|)-\frac{7}{x}$$

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.

$$x^{5}-x^{3}+10\ln(|x|)-\frac{7}{x}+С$$

Differentiate w.r.t. x

$$\frac{5x^{6}-3x^{4}+10x+7}{x^{2}}$$