Tough DI

Ans) I = | ₁₀ ∫¹⁹ (sin x / 1+x8 ) dx | ≤ ₁₀∫¹⁹ (sin x / 1+x8 ) dx

(Using property | _a ∫ ^b f(x) dx | ≤ _a ∫ ^b f(x) dx

Since |sinx| ≤1 for x≥10

Therefore, |sinx / (1+x ⁸) | ≤ 1 / |(1+x ⁸) | ...........(1)
10 ≤ x ≤ 19
so, 1+x⁸ > 10 ⁸
1 / (1+x⁸) < 1/ 10 ⁸ or
1 / |1+x ⁸| < 10 ^-8 .........................(2)

From (1) and (2),
|sinx / (1+x ⁸) | ≤ 10 ^-8 is fullfilled
| ₁₀∫¹⁹ (sin x / 1+x⁸ ) dx | < ₁₀∫¹⁹ 10^-8 dx

So, |₁₀∫¹⁹ (sin x / 1+x⁸ ) dx | < (19-10) 10 ^-8 < 10^-7
Therefore, the true value of the integral = 10 ^-8

Tushar you have got the ans..saw your working its.its (D)

₁₀∫¹⁹ (sin x / 1+x⁸ ) dx < 10^-7

The true value of the integral ≈ 10 ^-8

I found same problem solved by I.A.Maron...

Same as yours but summarized..im posting that below.

Since |sinx| ≤1 for x≥10

|sinx / (1+x ⁸) | ≤ 10 -8 is fulfilled

So, |₁₀∫¹⁹ (sin x / 1+x⁸ ) dx | < (19-10) 10 ^-8 < 10^-7

( The true value of the integral ≈ 10 ^-8 )