Solve ∫ (-x^2-4x-2)(-3x-7) wrt x

$$\int ( - x ^ { 2 } - 4 x - 2 ) ( - 3 x - 7 ) d x$$

Evaluate

$$\frac{3x^{4}}{4}+\frac{19x^{3}}{3}+17x^{2}+14x+С$$

Solution Steps

Use the distributive property to multiply -x^{2}-4x-2 by -3x-7 and combine like terms.

$$\int -3\left(-x^{2}\right)x-7\left(-x^{2}\right)+12x^{2}+34x+14\mathrm{d}x$$

Multiply -3 and -1 to get 3.

$$\int 3x^{2}x-7\left(-x^{2}\right)+12x^{2}+34x+14\mathrm{d}x$$

To multiply powers of the same base, add their exponents. Add 2 and 1 to get 3.

$$\int 3x^{3}-7\left(-x^{2}\right)+12x^{2}+34x+14\mathrm{d}x$$

Multiply -7 and -1 to get 7.

$$\int 3x^{3}+7x^{2}+12x^{2}+34x+14\mathrm{d}x$$

Combine 7x^{2} and 12x^{2} to get 19x^{2}.

$$\int 3x^{3}+19x^{2}+34x+14\mathrm{d}x$$

Integrate the sum term by term.

$$\int 3x^{3}\mathrm{d}x+\int 19x^{2}\mathrm{d}x+\int 34x\mathrm{d}x+\int 14\mathrm{d}x$$

Factor out the constant in each of the terms.

$$3\int x^{3}\mathrm{d}x+19\int x^{2}\mathrm{d}x+34\int x\mathrm{d}x+\int 14\mathrm{d}x$$

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 3 times \frac{x^{4}}{4}.

$$\frac{3x^{4}}{4}+19\int x^{2}\mathrm{d}x+34\int x\mathrm{d}x+\int 14\mathrm{d}x$$

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

$$\frac{3x^{4}}{4}+\frac{19x^{3}}{3}+34\int x\mathrm{d}x+\int 14\mathrm{d}x$$

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

$$\frac{3x^{4}}{4}+\frac{19x^{3}}{3}+17x^{2}+\int 14\mathrm{d}x$$

Find the integral of 14 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{3x^{4}}{4}+\frac{19x^{3}}{3}+17x^{2}+14x$$

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.

$$\frac{3x^{4}}{4}+\frac{19x^{3}}{3}+17x^{2}+14x+С$$

Differentiate w.r.t. x

$$\left(3x+7\right)\left(x^{2}+4x+2\right)$$