Solve ∫ (2x-5)(3x+1) wrt x

$$\int ( 2 x - 5 ) ( 3 x + 1 ) d x$$

Evaluate

$$2x^{3}-\frac{13x^{2}}{2}-5x+С$$

Solution Steps

Apply the distributive property by multiplying each term of 2x-5 by each term of 3x+1.

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

Combine 2x and -15x to get -13x.

$$\int 6x^{2}-13x-5\mathrm{d}x$$

Integrate the sum term by term.

$$\int 6x^{2}\mathrm{d}x+\int -13x\mathrm{d}x+\int -5\mathrm{d}x$$

Factor out the constant in each of the terms.

$$6\int x^{2}\mathrm{d}x-13\int x\mathrm{d}x+\int -5\mathrm{d}x$$

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

$$2x^{3}-13\int x\mathrm{d}x+\int -5\mathrm{d}x$$

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 -13 times \frac{x^{2}}{2}.

$$2x^{3}-\frac{13x^{2}}{2}+\int -5\mathrm{d}x$$

Find the integral of -5 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$2x^{3}-\frac{13x^{2}}{2}-5x$$

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.

$$2x^{3}-\frac{13x^{2}}{2}-5x+С$$

Differentiate w.r.t. x

$$\left(2x-5\right)\left(3x+1\right)$$