evaluate the integral using integration by parts.\n\\(\\int x^{2}\\cos(\\frac{1}{6}x)dx\\)\n\nb…
evaluate the integral using integration by parts.\n\\(\\int x^{2}\\cos(\\frac{1}{6}x)dx\\)\n\nb. \\(6x^{2}\\sin(\\frac{1}{6}x)+\\int 12x\\sin(\\frac{1}{6}x)dx\\)\n\nc. \\(6x^{2}\\sin(\\frac{1}{6}x)-\\int 12x\\sin(\\frac{1}{6}x)dx\\)\n\nd. \\(12x\\sin(\\frac{1}{6}x)+\\int 6x^{2}\\sin(\\frac{1}{6}x)dx\\)\n\nevaluate the integral.\n\\(\\int x^{2}\\cos(\\frac{1}{6}x)dx = \\square\\)
Answer
Explanation:
Step1: Apply integration - by - parts formula
The integration - by - parts formula is $\int u\mathrm{d}v=uv-\int v\mathrm{d}u$. Let $u = x^{2}$ and $\mathrm{d}v=\cos(\frac{1}{6}x)\mathrm{d}x$. First, find $\mathrm{d}u$ and $v$. Differentiating $u = x^{2}$ gives $\mathrm{d}u = 2x\mathrm{d}x$. Integrating $\mathrm{d}v=\cos(\frac{1}{6}x)\mathrm{d}x$, we have $v = 6\sin(\frac{1}{6}x)$ (since $\int\cos(ax)\mathrm{d}x=\frac{1}{a}\sin(ax)+C$, here $a=\frac{1}{6}$).
Step2: Substitute into the formula
$\int x^{2}\cos(\frac{1}{6}x)\mathrm{d}x=x^{2}\times6\sin(\frac{1}{6}x)-\int6\sin(\frac{1}{6}x)\times2x\mathrm{d}x=6x^{2}\sin(\frac{1}{6}x)-\int12x\sin(\frac{1}{6}x)\mathrm{d}x$
Answer:
C. $6x^{2}\sin(\frac{1}{6}x)-\int12x\sin(\frac{1}{6}x)\mathrm{d}x$