Use geometry and properties of integrals to evaluate the following definite inte

Sinead Mcgee 2021-10-19 Answered
Use geometry and properties of integrals to evaluate the following definite integrals.
40(2x+16x2)dx
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Expert Answer

nitruraviX
Answered 2021-10-20 Author has 101 answers

Given:
40(2x+16x2) dx 
Let x2=u
Then 2x dx =dux dx =12du
With these substitutions, the integral in (1) transforms to
I=12sin(u)cos8(u)du (2)
Now the integrand in (2)
Suggests that cosine function should be used for change of variable,
Since its derivative sine function is present in the integrand. Thus
Let cos(u)=v
Then sin(u)du=dvsin(u)du=dv
Therefore
I=12v8dv (3)
now,
We know that for n1,xn+1n+1 is an antiderivative of the function f(x)=xn
So that
I=12[v99]+C
=118cos9(u)+C
=118cos9(x2)+C

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