Question

Show that |∫_−2𝜋^2𝜋 𝑥 ² sin⁸ (𝑒 ^𝑥 ) 𝑑𝑥| ≤ (16𝜋 ³ )/3

Answer 1

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:

$|\int^{2\pi}_{-2\pi} x^2sin^8(e^x)dx|\le \int^{2\pi}_{-2\pi} |x^2sin^8(e^x)|dx=$

$=\int^{2\pi}_{-2\pi} |x^2||sin^8(e^x)|dx\le \int^{2\pi}_{-2\pi} |x^2|dx=\int^{2\pi}_{-2\pi} x^2dx=\frac{16\pi^3}{3}$