Show that |∫−2𝜋2𝜋 𝑥 2 sin8 (𝑒 𝑥 ) 𝑑𝑥| ≤ (16𝜋 3 )/3
If f is integrable on [a,b], then |f| is integrable on [a,b] and
"|\\int^b_a f(x)dx|\\le \\int^b_a |f|dx"
So:
"|\\int^{2\\pi}_{-2\\pi} x^2sin^8(e^x)dx|\\le \\int^{2\\pi}_{-2\\pi} |x^2sin^8(e^x)|dx="
"=\\int^{2\\pi}_{-2\\pi} |x^2||sin^8(e^x)|dx\\le \\int^{2\\pi}_{-2\\pi} |x^2|dx=\\int^{2\\pi}_{-2\\pi} x^2dx=\\frac{16\\pi^3}{3}"
Comments
Leave a comment