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

Find the integral. ((2x + 47? ax x + 8x? 3 + 8x2 + 16x + c O 4x3 + 16x2 + 4x + c 0 8x-example-1
User Mchicago
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1 Answer

15 votes
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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=(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(x^3)/(3)=(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(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(x)/(1)=16x

Then, by adding all results we get


\int (2x+4)^2dx=(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=(4)/(3)x^3+8x^2+16x+C

User Nikolay Tomitov
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