Solve ∫ left(5x+2right)^2left(8-xright)^2 wrt x

$$\int{ { \left(5x+2 \right) }^{ 2 } { \left(8-x \right) }^{ 2 } }d x$$

Evaluate

$$5x^{5}-95x^{4}+428x^{3}+608x^{2}+256x+С$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5x+2\right)^{2}.

$$\int \left(25x^{2}+20x+4\right)\left(8-x\right)^{2}\mathrm{d}x$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(8-x\right)^{2}.

$$\int \left(25x^{2}+20x+4\right)\left(64-16x+x^{2}\right)\mathrm{d}x$$

Use the distributive property to multiply 25x^{2}+20x+4 by 64-16x+x^{2} and combine like terms.

$$\int 1284x^{2}-380x^{3}+25x^{4}+1216x+256\mathrm{d}x$$

Integrate the sum term by term.

$$\int 1284x^{2}\mathrm{d}x+\int -380x^{3}\mathrm{d}x+\int 25x^{4}\mathrm{d}x+\int 1216x\mathrm{d}x+\int 256\mathrm{d}x$$

Factor out the constant in each of the terms.

$$1284\int x^{2}\mathrm{d}x-380\int x^{3}\mathrm{d}x+25\int x^{4}\mathrm{d}x+1216\int x\mathrm{d}x+\int 256\mathrm{d}x$$

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 1284 times \frac{x^{3}}{3}.

$$428x^{3}-380\int x^{3}\mathrm{d}x+25\int x^{4}\mathrm{d}x+1216\int x\mathrm{d}x+\int 256\mathrm{d}x$$

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 -380 times \frac{x^{4}}{4}.

$$428x^{3}-95x^{4}+25\int x^{4}\mathrm{d}x+1216\int x\mathrm{d}x+\int 256\mathrm{d}x$$

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 25 times \frac{x^{5}}{5}.

$$428x^{3}-95x^{4}+5x^{5}+1216\int x\mathrm{d}x+\int 256\mathrm{d}x$$

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 1216 times \frac{x^{2}}{2}.

$$428x^{3}-95x^{4}+5x^{5}+608x^{2}+\int 256\mathrm{d}x$$

Find the integral of 256 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$428x^{3}-95x^{4}+5x^{5}+608x^{2}+256x$$

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.

$$428x^{3}+608x^{2}+256x-95x^{4}+5x^{5}+С$$

Differentiate w.r.t. x

$$\left(\left(x-8\right)\left(5x+2\right)\right)^{2}$$