{
  "problem": "Evaluate $~~\\int^{2\\pi}_\\pi x\\sin\\Big(\\dfrac x2\\Big) dx\\,$. Choose 1 answer: Choose 1 answer: (Choice A) A $0$ (Choice B) B $2\\pi-4$ (Choice C) C $4\\pi-4$ (Choice D) D $6\\pi$",
  "hints": [
    "We will solve this by integrating by parts. We know that $ \\int u(x)v\\,^\\prime(x)dx = u(x)v(x)-\\int u\\,^\\prime(x)v(x)dx\\,$. We can rewrite this as $ \\int u\\ dv = uv-\\int v\\ du\\,$.",
    "In this problem we will let $~u = x~$ and $~dv=\\sin\\Big(\\dfrac x2\\Big)\\, dx\\,$.",
    "Then $~du = dx~$ and $~v = \\int \\sin\\Big(\\dfrac x2\\Big)\\,dx = -2\\,{\\cos\\Big(\\dfrac x2\\Big)}$.",
    "Integration by parts gives $ \\int^{2\\pi}_\\pi x\\sin\\Big(\\dfrac x2\\Big) dx= x\\cdot\\Bigg(-2\\cos\\Big(\\dfrac x2\\Big)\\Bigg)\\Bigg]_\\pi^{2\\pi}-\\int^{2\\pi}_\\pi-2\\cos\\Big(\\dfrac x2\\Big)dx$ $ ~~~~~= -2x\\,\\cos\\Big(\\dfrac x2\\Big)+4\\sin\\Big(\\dfrac x2\\Big)\\Bigg]^{2\\pi}_\\pi$ $ ~~~~~=\\Big(4\\pi+0\\Big)-\\Big(0+4\\Big) = 4\\pi-4\\,$."
  ]
}
{'source': 'AMPS', 'file': 'amps/khan/integration-by-parts--definite-integrals/16075985833.json'}