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