Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, give the answer -1. ∫4∞xe−2x

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Sally Cresswell · Accepted Answer

Compute the definite integral: ∫4∞e−2xxdx For the integrand e−2xx, integrate by parts, ∫fdg=fg−∫gdf, where f=x,dg=e−2xdx,df=dx,g=−12e−2x: =(−12e−2xx)|4∞+12∫4∞e−2xdx INTERMEDIATE STEPS: Possible derivation: ddx(x) Use the power rule, ddx(xn)=nxn−1, where n = 1. ddx(x)=ddx(x1)=x0: Answer: = 1 Evaluate the antiderivative at the limits and subtract. (−12e−2xx)|4∞=(limb→∞−12e−2bb)−(−12e−244)=(limb→∞−12e−2bb)−(−2e8): =(limb→∞−12e−2bb)+2e8+12∫4∞e−2xdx limb→∞−12e−2bb=0: =2e8+12∫4∞e−2xdx INTERMEDIATE STEPS: Find the following limit: limx→∞−12e−2xx Hint: Factor a constant multiple out of the limit. limx→∞−12xe−2x=−12limx→∞xe−2x: −limx→∞e−2xx2 Hint: | Linear functions grow asymptotically slower than exponential functions. Since the polynomial x grows asymptotically slower than e2x as x approaches ∞,

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

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