Solve ∫ (sinx-2x^3+4/x) wrt x

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

Evaluate

$$4\ln(|x|)-\cos(x)-\frac{x^{4}}{2}+С$$

Solution Steps

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

Use \int \sin(x)\mathrm{d}x=-\cos(x) from the table of common integrals to obtain the result.

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

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}. Multiply -2 times \frac{x^{4}}{4}.

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

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

$$-\cos(x)-\frac{x^{4}}{2}+4\ln(|x|)$$

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.

$$-\cos(x)-\frac{x^{4}}{2}+4\ln(|x|)+С$$

Differentiate w.r.t. x

$$\sin(x)-2x^{3}+\frac{4}{x}$$