Question

Find the integral. ((2x + 47? ax x + 8x? 3 + 8x2 + 16x + c O 4x3 + 16x2 + 4x + c 0 8x + 16 + C x + 4x + c

Answer 1

To solve this problem, we will use the following properties

$\int f+gdx\text{ =}\int fdx+\int gdx$

(The integral of the sum is the sum of the integrals).

$\int cfdx=c\int fdx$

(The integral of a constant times a function is constant times the integral of the function)

$\int x^ndx=\frac{x^{n+1}}{n+1}^{}$

only when n is not -1.

First, recall that

$\int (2x+4)^2dx=\int 4x^2+16x+16dx$

using the sum property, we have

$\int 4x^2+16x+16dx=\int 4x^2dx+\int 16xdx+\int 16dx$

Now, we will solve each integral apart. We will use the last two properties to do so.

Note that

$\int 4x^2=4\int x^2$

we identify that in this case, the function is of the form x^n where n=2. Then, using the last property we get

$\int 4x^2dx=4\frac{x^3}{3}=\frac{4}{3}x^3^{}$

Also, note that

$\int 16xdx=16\int xdx$

we identify that the function is of the form x^n where n=1. Then,

$\int 16xdx=16\frac{x^2}{2}=8x^2$

Finally, we have

$\int 16dx=16\cdot\int 1dx$

we note that 1=x⁰. So we have

$\int 16dx=16\cdot\frac{x}{1}=16x$

Then, by adding all results we get

$\int (2x+4)^2dx=\frac{4}{3}x^3+8x^2+16x$

Since we are finding the function whose derivative is (2x+4)², we add a constant C. So we have

$\int (2x+4)^2dx=\frac{4}{3}x^3+8x^2+16x+C$