Solve ∫ (3x^4-2x+1/x) wrt x

$$\int ( 3 x ^ { 4 } - 2 x + \frac { 1 } { x } ) d x$$

Evaluate

$$\ln(|x|)+\frac{3x^{5}}{5}-x^{2}+С$$

Solution Steps

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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 3 times \frac{x^{5}}{5}.

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

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}. Multiply -2 times \frac{x^{2}}{2}.

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

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

$$\frac{3x^{5}}{5}-x^{2}+\ln(|x|)$$

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.

$$\frac{3x^{5}}{5}-x^{2}+\ln(|x|)+С$$

Differentiate w.r.t. x

$$3x^{4}-2x+\frac{1}{x}$$