Find the integral?

Question

Find the integral of 2^5x dx from negative infinity to 0.

When I integrate, I got 2^(5x)/(5*ln(2)) from negative infinity to 0, and I got 1/(5*ln(2))+infinity, but the answer should be positive infinity, why is the answer in the book says 1/(5*ln(2))? What's the correct answer? Show work.

 

Answer 1

∫_-∞⁰ 2^5xdx= 1/5∫_-∞⁰2^zdz=1/5(2^z/ln(2))|_-∞⁰=1/(5*ln(2))[2⁰-lim_z→-∞2^z]=1/(5*ln(2))[1-0]=1/(5*ln(2))

 

Here I made a change of variable z=5x. Lim 2^z when z goes to - infinity is obviously zero.

Answer 2

∫{-∞, 0}2^(5x) dx

= (1/5)∫{-∞, 0}2^(5x) d5x

= (1/5)2^(5x)/ln2 from -∞ to 0

= 1/(5ln2) <==Answer

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Attn: (1/5)2^(5x)/ln2 at x = -∞ is equal to zero.