|∫(1 to root3) {[e^(-x) sinx]/x^2 +1[dx]}| ≤ {π/12e}
prove true or false and justfy
In the interval 1 to √3, e^(-x) is positive.
In the interval 1 to √3, sin(x) is positive
In the interval 1 to √3, x^2 is positive
Therefore in the interval 1 to √3, the quantity e^(-x).sin(x)/x^2 is positive.
Since e^(-x).sin(x)/x^2 is always positive in the given interval then the area under the curve will also be positive.
i.e. |∫(1 to √3) {[e^(-x) sinx]/x^2 [dx]}| is positive
Let |∫(1 to √3) {[e^(-x) sinx]/x^2 [dx]}| = δ, a positive number.
Then |∫(1 to √3) {[e^(-x) sinx]/x^2 + 1[dx]}| = |∫(1 to √3) {[e^(-x) sinx]/x^2 [dx]}| + |∫(1 to √3) {[1 [dx]}|
|∫(1 to √3) {[e^(-x) sinx]/x^2 + 1[dx] = δ + |∫(1 to √3) {[1 [dx]}|
|∫(1 to √3) {[e^(-x) sinx]/x^2 + 1[dx] = δ + (√3 – 1)
i.e. δ + (√3 – 1) ≤ (π/12)e
δ ≤ (π/12)e – (√3 – 1) = 0.71165 – 0.73205 = -0.0204
δ ≤ -0.0204
But δ is a positive quantity
Therefore, the statement is false.