Solve ∫ (2x^2-7x+3)(-3x^2-2x-5) wrt x

$$\int ( 2 x ^ { 2 } - 7 x + 3 ) ( - 3 x ^ { 2 } - 2 x - 5 ) d x$$

Evaluate

$$-\frac{6x^{5}}{5}+\frac{17x^{4}}{4}-\frac{5x^{3}}{3}+\frac{29x^{2}}{2}-15x+С$$

Solution Steps

Use the distributive property to multiply 2x^{2}-7x+3 by -3x^{2}-2x-5 and combine like terms.

$$\int -6x^{4}+17x^{3}-5x^{2}+29x-15\mathrm{d}x$$

Integrate the sum term by term.

$$\int -6x^{4}\mathrm{d}x+\int 17x^{3}\mathrm{d}x+\int -5x^{2}\mathrm{d}x+\int 29x\mathrm{d}x+\int -15\mathrm{d}x$$

Factor out the constant in each of the terms.

$$-6\int x^{4}\mathrm{d}x+17\int x^{3}\mathrm{d}x-5\int x^{2}\mathrm{d}x+29\int x\mathrm{d}x+\int -15\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{4}\mathrm{d}x with \frac{x^{5}}{5}. Multiply -6 times \frac{x^{5}}{5}.

$$-\frac{6x^{5}}{5}+17\int x^{3}\mathrm{d}x-5\int x^{2}\mathrm{d}x+29\int x\mathrm{d}x+\int -15\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{3}\mathrm{d}x with \frac{x^{4}}{4}. Multiply 17 times \frac{x^{4}}{4}.

$$-\frac{6x^{5}}{5}+\frac{17x^{4}}{4}-5\int x^{2}\mathrm{d}x+29\int x\mathrm{d}x+\int -15\mathrm{d}x$$

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

$$-\frac{6x^{5}}{5}+\frac{17x^{4}}{4}-\frac{5x^{3}}{3}+29\int x\mathrm{d}x+\int -15\mathrm{d}x$$

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

$$-\frac{6x^{5}}{5}+\frac{17x^{4}}{4}-\frac{5x^{3}}{3}+\frac{29x^{2}}{2}+\int -15\mathrm{d}x$$

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

$$-\frac{6x^{5}}{5}+\frac{17x^{4}}{4}-\frac{5x^{3}}{3}+\frac{29x^{2}}{2}-15x$$

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.

$$-\frac{6x^{5}}{5}+\frac{17x^{4}}{4}-\frac{5x^{3}}{3}+\frac{29x^{2}}{2}-15x+С$$

Differentiate w.r.t. x

$$\left(x-3\right)\left(2x-1\right)\left(-3x^{2}-2x-5\right)$$