Question

`int_0^pi x*sin x*cos^4x  "d"x` = ______.

पर्याय

• `pi/4`

• `pi/5`

• `pi/10`

• `pi/20`

Answer 1

`int_0^pi x*sin x*cos^4x  "d"x` = `pi/5`.

Explanation:

Let I = `int_0^pi x*sin x*cos^4x  "d"x`  ......(i)

∴ I = `int_0^pi (pi - x)sin(pi - x)cos^4(pi - x)  "d"x` ......`[because int_0^"a" "f"(x)"d"x = int_0^"a" "f"("a" - x)"d"x]`

∴ I = `int_0^"a"(pi - x)sin x cos^4x  "d"x` ......(ii)

Adding (i) and (ii), we get

2I = `pi int_0^"a" sinx*cos^4x  "d"x`

Put cos x = t ⇒ – sin x dx = dt

∴ 2I = `- pi int_1^(-1) "t"^4 "dt"`

∴ I = `pi/2 int_(-1)^1 "t"^4 "dt"` ......`[because int_"a"^"b" "f"(x)"d"x = -int_"b"^"a" "f"(x)"d"x]`

= `2(pi/2) int_0^1 "t"^4 "dt"` ......[∵ t⁴ is an even function]

= `pi["t"^5/5]_0^1`

= `pi(1/5 - 0)`

∴ I = `pi/5`