Question

If `∫ x^3 sin^4 (x^4) cos (x^4) dx = a sin^5 (x^4) + C`, then a is equal to ______.

Options

• `-1/10`

• `1/20`

• `1/4`

• `1/5`

Answer 1

If `∫ x^3 sin^4 (x^4) cos (x^4) dx = a sin^5 (x^4) + C`, then a is equal to `underlinebb(1/20)`.

Explanation:

Finding `∫ x^3 sin^4 (x^4) cos (x^4) dx`

Let t = sin (x⁴)

Differentiating t:

`dt/dx` = cos (x⁴) × (x⁴)

`dt/dx` = cos (x⁴) × 4x³

dt = cos (x⁴) 4x³ dx

`dt/4` = cos (x⁴) x³ dx

Now, 

`∫ x^3 sin^4 (x^4) cos (x^4) dx = int(t^4dt)/4`

= `1/4 int t^4 dt`

= `1/4 xx t^5/5 + C`

= `1/20 t^5 + C`

Putting back t = sin (x⁴)

= `1/20 sin^5 (x^4) + C`

Comparing with a sin⁵ (x⁴) + C, then a is equal to `1/20`.