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Solve the following : ∫0911+x⋅dx - Mathematics and Statistics

Sum

Solve the following : `int_0^9 (1)/(1 + sqrt(x))*dx`

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Solution

Let I = `int_0^9 (1)/(1 + sqrt(x))*dx`
Put `1 + sqrt(x)` = t
∴ x = (t – 1)2
∴ dx = 2(t – 1)dt
When x = 0, t = 1 + 0 = 1
When x = 9, t = `1 + sqrt(9)`
= 1 + 3 = 4

∴ I = `int_1^4 (2(t - 1))/"t"*"dt"`

= `2int_1^4(1 - 1/"t")*"dt"`

= `2]"t" - log|"t"|]_1^4`
= 2 [(4 – log |4|) – (1 – log |1|)]
= 2 [4 – log 4 – (1 – 0)]
= 2 [4 – log 22 – 1)
= 2 (3 – 2log 2)
∴ I = 6 – 4 log 2.

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APPEARS IN

Balbharati Mathematics and Statistics 1 (Commerce) 12th Standard HSC Maharashtra State Board
Chapter 6 Definite Integration
Miscellaneous Exercise 6 | Q 4.2 | Page 150
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