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If eeeed∫3ex-5e-x4e6x+5e-xdx = ax + b log |4ex + 5e –x| + C, then ______. - Mathematics

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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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Definite Integrals
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Q 22
Chapter 7: Integrals - Solved Examples [Page 159]

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NCERT Exemplar Mathematics [English] Class 12
Chapter 7 Integrals
Solved Examples | Q 22 | Page 159

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