Project ForgeEngineering Mathematics
Calculus
GATE 2018
Hard
The value of ∫[0 to π/4] x·cos(x²) dx correct to three decimal places (assuming that π = 3.14) is ______.
0.289(± 0.015 accepted)
Explanation
Substitute u=x², du=2x dx, so x dx = du/2. When x=0, u=0; when x=π/4, u=π²/16. The integral becomes (1/2)∫[0 to π²/16] cos(u) du = (1/2)sin(π²/16). With π=3.14, π²/16 ≈ 0.6162 radians, and sin(0.6162) ≈ 0.578, giving (1/2)(0.578) ≈ 0.289 — matching the file's own answer-key range (0.27 to 0.30).
Source: GATE CS 2018 (2018)
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