Solve ∫ (2x+5)(x-3) wrt x

$$\int ( 2 x + 5 ) ( x - 3 ) d x$$

Evaluate

$$\frac{2x^{3}}{3}-\frac{x^{2}}{2}-15x+С$$

Solution Steps

Apply the distributive property by multiplying each term of 2x+5 by each term of x-3.

$$\int 2x^{2}-6x+5x-15\mathrm{d}x$$

Combine -6x and 5x to get -x.

$$\int 2x^{2}-x-15\mathrm{d}x$$

Integrate the sum term by term.

$$\int 2x^{2}\mathrm{d}x+\int -x\mathrm{d}x+\int -15\mathrm{d}x$$

Factor out the constant in each of the terms.

$$2\int x^{2}\mathrm{d}x-\int x\mathrm{d}x+\int -15\mathrm{d}x$$

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

$$\frac{2x^{3}}{3}-\int x\mathrm{d}x+\int -15\mathrm{d}x$$

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 -1 times \frac{x^{2}}{2}.

$$\frac{2x^{3}}{3}-\frac{x^{2}}{2}+\int -15\mathrm{d}x$$

Find the integral of -15 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{2x^{3}}{3}-\frac{x^{2}}{2}-15x$$

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.

$$\frac{2x^{3}}{3}-\frac{x^{2}}{2}-15x+С$$

Differentiate w.r.t. x

$$\left(x-3\right)\left(2x+5\right)$$