Find $\int x^2 \cos n x \,\mathrm{d}x$, where $n$ is a positive integer.[3] Hence find \(\i

Question

Find $\int x^2 \cos n x \,\mathrm{d}x$, where $n$ is a positive integer.[3] Hence find $\int_{\pi}^{2\pi} x^2 \cos n x \,\mathrm{d}x$, giving your answers in the form $a\frac{\pi}{n^2}$, where the possible values of $a$ are to be determined.[2] The region bounded by the curve $y = \frac{x\sqrt{x}}{9-x^2}$, the $x$-axis and the lines $x = 0$ and $x = 2$ is rotated through $2\pi$ radians about the $x$-axis. Use the substitution $u = 9 - x^2$ to find the exact volume of the solid obtained, simplifying your answer.[5]

Accepted Answer

(a)(i) $\dfrac{x^2\sin(nx)}n+\dfrac{2x\cos(nx)}{n^2}-\dfrac{2\sin(nx)}{n^3}+C$ (ii) $\dfrac{6\pi}{n^2}$ if $n$ is odd and $\dfrac{2\pi}{n^2}$ if $n$ is even; $a=6$ or $2$ (b) $\dfrac\pi2\left(\dfrac45+\ln\dfrac59\right)\text{ units}^3$

N2016 P2 Q2

N2016 P2 Q2

Video Solution:

Solution 1