Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, give the answer -1.

Jaya Legge 2021-09-02 Answered

Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, give the answer -1.
4xe2x

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Expert Answer

Sally Cresswell
Answered 2021-09-03 Author has 91 answers

Compute the definite integral:
4e2xxdx
For the integrand e2xx, integrate by parts, fdg=fggdf, where
f=x,dg=e2xdx,df=dx,g=12e2x:
=(12e2xx)|4+124e2xdx
INTERMEDIATE STEPS:
Possible derivation:
ddx(x)
Use the power rule, ddx(xn)=nxn1, where n = 1.
ddx(x)=ddx(x1)=x0:
Answer: = 1
Evaluate the antiderivative at the limits and subtract.
(12e2xx)|4=(limb12e2bb)(12e244)=(limb12e2bb)(2e8):
=(limb12e2bb)+2e8+124e2xdx
limb12e2bb=0:
=2e8+124e2xdx
INTERMEDIATE STEPS:
Find the following limit:
limx12e2xx
Hint: Factor a constant multiple out of the limit.
limx12xe2x=12limxxe2x:
limxe2xx2
Hint: | Linear functions grow asymptotically slower than exponential functions.
Since the polynomial x grows asymptotically slower than e2x as x approaches ∞,

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