Solve ∫ (x+1)(2x-1)(x+9) wrt x

$$\int ( x + 1 ) ( 2 x - 1 ) ( x + 9 ) d x$$

Evaluate

$$\frac{x^{4}}{2}+\frac{19x^{3}}{3}+4x^{2}-9x+С$$

Solution Steps

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

$$\int \left(2x^{2}-x+2x-1\right)\left(x+9\right)\mathrm{d}x$$

Combine -x and 2x to get x.

$$\int \left(2x^{2}+x-1\right)\left(x+9\right)\mathrm{d}x$$

Apply the distributive property by multiplying each term of 2x^{2}+x-1 by each term of x+9.

$$\int 2x^{3}+18x^{2}+x^{2}+9x-x-9\mathrm{d}x$$

Combine 18x^{2} and x^{2} to get 19x^{2}.

$$\int 2x^{3}+19x^{2}+9x-x-9\mathrm{d}x$$

Combine 9x and -x to get 8x.

$$\int 2x^{3}+19x^{2}+8x-9\mathrm{d}x$$

Integrate the sum term by term.

$$\int 2x^{3}\mathrm{d}x+\int 19x^{2}\mathrm{d}x+\int 8x\mathrm{d}x+\int -9\mathrm{d}x$$

Factor out the constant in each of the terms.

$$2\int x^{3}\mathrm{d}x+19\int x^{2}\mathrm{d}x+8\int x\mathrm{d}x+\int -9\mathrm{d}x$$

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

$$\frac{x^{4}}{2}+19\int x^{2}\mathrm{d}x+8\int x\mathrm{d}x+\int -9\mathrm{d}x$$

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}.

$$\frac{x^{4}}{2}+\frac{19x^{3}}{3}+8\int x\mathrm{d}x+\int -9\mathrm{d}x$$

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

$$\frac{x^{4}}{2}+\frac{19x^{3}}{3}+4x^{2}+\int -9\mathrm{d}x$$

Find the integral of -9 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x^{4}}{2}+\frac{19x^{3}}{3}+4x^{2}-9x$$

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.

$$\frac{x^{4}}{2}+\frac{19x^{3}}{3}+4x^{2}-9x+С$$

Differentiate w.r.t. x

$$\left(2x-1\right)\left(x+1\right)\left(x+9\right)$$