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प्रश्न
Evaluate the following:
`int_0^(2pi) sin^7 x/4 "d"x`
उत्तर
Put t = `x/4`
dt = `("d"x)/4`
4 dt = dx
| x | 0 | `2pi` |
| t | 0 | `pi/2` |
`int_0^(2pi) sin^7 x/4 "d"x = 4 int_0^(pi/2) sin^7 "t" "dt"`
Here n = 7, which is odd
∴ `int_0^(pi/2) sin^"n" x "d"x = (("n" - 1))/"n" xx (("n" - 3))/(("n" - 2)) xx ...... xx 2/3`
= `4 xx 6/7 xx 4/5 xx 2/3 xx 1`
= `64/35`
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