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Question
If `int (3"e"^x - 5"e"^-x)/(4"e"6x + 5"e"^-x)"d"x` = ax + b log |4ex + 5e –x| + C, then ______.
Options
a = `(-1)/8`, b = `7/8`
a = `1/8`, b = `7/8`
a = `(-1)/8`, b = `(-7)/8`
a = `1/8`, b = `(-7)/8`
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Solution
If `int (3"e"^x - 5"e"^-x)/(4"e"6x + 5"e"^-x)"d"x` = ax + b log |4ex + 5e –x| + C, then a = `(-1)/8`, b = `(-7)/8`.
Explanation:
`(3"e"^x - 5"e"^-x)/(4"e"^x + 5"e"^-x) = "a" + "b" ((4"e"^x - 5"e"^-x))/(4"e"^x + 5"e"^-x)`,
Giving 3ex – 5e –x = a(4ex + 5e–x) + b(4ex – 5e–x).
Comparing coefficients on both sides,
We get 3 = 4a + 4b and –5 = 5a – 5b.
This verifies a = `(-1)/8`, b = `7/8`.
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