Question

How do you find the integral of `int 48((sin(sqrt(x))^3)/(sqrt(x))`?

Answer 1

`4\cos(3\sqrtx)-36\cos (\sqrtx)+C`

Explanation:

Assuming there is a typo in the question, it can possibly taken as

`\int 48(\frac{(\sin(\sqrtx))^3}{\sqrtx})\ dx`

`=48\int\frac{\sin^3(\sqrtx)}{\sqrtx}\ dx`

Let `sqrtx=t\implies 1/{2\sqrtx}\ dx=dt\ \ or\ \ dx=2t dt`

`=48\int\frac{\sin^3t}{t}(2t\ dt)`

`=48\int \sin^3t\ dt`

`=48\int\frac{3\sint-\sin(3t)}{4}\ dt`

`=12(3\int \sin t\ dt-\int \sin(3t)\ dt)`

`=12(3(-\cos t)- \frac{-\cos(3t)}{3}) +C`

`=4\cos(3t)-36\cos t+C`

`=4\cos(3\sqrtx)-36\cos (\sqrtx)+C`