Find \[ \left( 6\sqrt{x-4x^{2}} + 5 \right)dx - Scottish Highers Maths - Question 2 - 2019

Next, we simplify and integrate:

$\int 6\sqrt{x - 4x^{2}} \,dx$

To do this, we can use substitution method or trigonometric identities, but considering the integrated variable complexity, we integrate directly to get:

$= 4\left(\frac{2}{3}x^{\frac{3}{2}}\right) = \frac{8}{3}x^{\frac{3}{2}}$

Now, we proceed to integrate the constant term, which is simply:

$\int 5 \,dx = 5x$

Combining both results, we have:

$\int \left(6\sqrt{x - 4x^{2}} + 5\right) dx = \frac{8}{3}x^{\frac{3}{2}} + 5x + C$

Therefore, the full solution is:

$\frac{8}{3}x^{\frac{3}{2}} + 5x + C$