How to know what to multiply inside and outside the integral in calculus?

Question

https://i.postimg.cc/XJ10J956/snip.png

How do we know to multiply by 2/3 and power of 3/2 in this screenshot?

Answer 1

When you are taking the integral, you have the equation in the integral

∫4π √(x+1)

That equation is equivalent to

4π∫(x+1)^(1/2)

We also know a few items also, 4π is a constant so it goes to the outside of the integral meaning you only integrate

(x+1)^(1/2)

When you integrate, you add 1 to the exponent giving you (x+1)^(3/2). But you know you have a 1 in front of the (x+1). Therefore, you need to solve for this.

(3/2)*C = 1

C is your coefficient that is in front which is 2/3.

Also, the answer that you have is wrong. You forgot to raise the inside of the square root by power of 3.

(8/3)*π*(√(3³) - √(2³) = (8/3)*π*(√(27) - √(8)