Mister Exam

# Integral of (x^2)*(cos(nx)) dx

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### The solution

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|   2
|  x *cos(n*x) dx
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$$\int\limits_{0}^{1} x^{2} \cos{\left(n x \right)}\, dx$$
Integral(x^2*cos(n*x), (x, 0, 1))
                          //                 3                           \
||                x                            |
||                --                  for n = 0|
||                3                            |
/                       ||                                             |
|                        ||/sin(n*x)   x*cos(n*x)                       |      //   x      for n = 0\
|  2                     |||-------- - ----------  for n != 0           |    2 ||                   |
| x *cos(n*x) dx = C - 2*|<|    2          n                            | + x *|

$$\int x^{2} \cos{\left(n x \right)}\, dx = C + x^{2} \left(\begin{cases} x & \text{for}\: n = 0 \\\frac{\sin{\left(n x \right)}}{n} & \text{otherwise} \end{cases}\right) - 2 \left(\begin{cases} \frac{x^{3}}{3} & \text{for}\: n = 0 \\\frac{\begin{cases} - \frac{x \cos{\left(n x \right)}}{n} + \frac{\sin{\left(n x \right)}}{n^{2}} & \text{for}\: n \neq 0 \\0 & \text{otherwise} \end{cases}}{n} & \text{otherwise} \end{cases}\right)$$
/sin(n)   2*sin(n)   2*cos(n)
|------ - -------- + --------  for And(n > -oo, n < oo, n != 0)
|  n          3          2
<            n          n
|
|            1/3                          otherwise
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$$\begin{cases} \frac{\sin{\left(n \right)}}{n} + \frac{2 \cos{\left(n \right)}}{n^{2}} - \frac{2 \sin{\left(n \right)}}{n^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\\frac{1}{3} & \text{otherwise} \end{cases}$$
=
=
/sin(n)   2*sin(n)   2*cos(n)
|------ - -------- + --------  for And(n > -oo, n < oo, n != 0)
|  n          3          2
<            n          n
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|            1/3                          otherwise
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$$\begin{cases} \frac{\sin{\left(n \right)}}{n} + \frac{2 \cos{\left(n \right)}}{n^{2}} - \frac{2 \sin{\left(n \right)}}{n^{3}} & \text{for}\: n > -\infty \wedge n < \infty \wedge n \neq 0 \\\frac{1}{3} & \text{otherwise} \end{cases}$$
Piecewise((sin(n)/n - 2*sin(n)/n^3 + 2*cos(n)/n^2, (n > -oo)∧(n < oo)∧(Ne(n, 0))), (1/3, True))

Use the examples entering the upper and lower limits of integration.

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