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Evaluate: ∫exe2x+4ex+13 dx - Mathematics

Sum

Evaluate: `int "e"^"x"/sqrt("e"^"2x" + 4"e"^"x" + 13)` dx

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Solution

Let I = `int "e"^"x"/sqrt("e"^"2x" + 4"e"^"x" + 13)` dx

`= int "e"^"x"/sqrt(("e"^"x")^2 +  4"e"^"x"  + 13)` dx

Put ex = t

∴ ex  dx = dt

∴ I = `"dt"/(sqrt("t"^2 + 4"t" + 13))`

`= int 1/sqrt("t"^2 + 4"t" + 4 - 4 + 13)` dt

`= int 1/(sqrt(("t + 2")^2 + 9))` dt

`= int 1/(sqrt(("t" + 2)^2 + (3)^2))` dt

`= log |"t" + 2 + sqrt(("t" + 2)^2 + (3)^2)|` + c

`= log |"t" + 2 + sqrt("t"^2 + 4"t" + 13)|`

∴ I = `log |"e"^"x" + 2 + sqrt("e"^"2x" + 4"e"^"x" + 13)|`

  Is there an error in this question or solution?
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APPEARS IN

Balbharati Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 5 Integration
Miscellaneous Exercise 5 | Q 4.3 | Page 139
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