The value of \(\int^{\pi/4} _0 x \cos(x^2) dx\) correct to three decimal…
2018
The value of \(\int^{\pi/4} _0 x \cos(x^2) dx\) correct to three decimal places
(assuming that \(𝜋 = 3.14\) ) is _____.
Attempted by 53 students.
Show answer & explanation
Correct answer: 0.2885 to 0.2895
Solution: Substitute u = x^2, so du = 2x dx and x dx = du/2.
The integral becomes (1/2) ∫ from u = 0 to u = (π/4)^2 of cos u du = (1/2)[sin u]0 to (π/4)^2 = (1/2)·sin((π/4)^2).
Using π = 3.14 as given, (π/4) = 0.785 and (π/4)^2 = 0.616225.
Compute sin(0.616225) ≈ 0.577965 (by calculator or a few terms of the Taylor series).
Thus the integral ≈ 0.5 × 0.577965 = 0.2889825, which to three decimal places is 0.289.
Final answer: 0.289