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प्रश्न
Evaluate the following integrals using properties of integration:
`int_0^pi sin^4 x cos^3 x "d"x`
उत्तर
`int_0^pi sin^4 x cos^3 x "d"x`
f(x) = sin4x cos3x
f(2π – x) = sin4(2π – x) cos3(2π – x)
= sin4x cos3x
f(2π – x) = f(x)
if `"f"(2"a" - x) = "f("x)` then `int_0^(2"a") "f"(x) "d"x`
= `2 int_0^"a" "f"(x) "d"x`
I = `2 int_0^pi sin^4 x cos^3 x "d"x`
t = sin x
dt = cox dx
| x | 0 | `pi` |
| t | 0 | 0 |
` 2int_0^pi sin^4x(1 - sin^2x) cosx "d"x`
Limit from 0 to π tends to 0 to 0
∴ Integral value = 0
∴ `int_0^pi sin^4 x cos^3x "d"x` = 0
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