Solve ∫ (2x-5y+2) wrt x

$$\int{ (2x-5y+2) }d x$$

Evaluate

$$x^{2}-5xy+2x+С$$

Solution Steps

Integrate the sum term by term.

$$\int 2x\mathrm{d}x+\int -5y\mathrm{d}x+\int 2\mathrm{d}x$$

Factor out the constant in each of the terms.

$$2\int x\mathrm{d}x-5\int y\mathrm{d}x+\int 2\mathrm{d}x$$

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}.

$$x^{2}-5\int y\mathrm{d}x+\int 2\mathrm{d}x$$

Find the integral of y using the table of common integrals rule \int a\mathrm{d}x=ax.

$$x^{2}-5yx+\int 2\mathrm{d}x$$

Find the integral of 2 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$x^{2}-5yx+2x$$

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.

$$x^{2}-5yx+2x+С$$

Differentiate w.r.t. x

$$2x-5y+2$$