Definite integrals

Question

Confused on the part where you insert numbers after the s (ex 1/2S u^3du)=u^4/8 how do I get the information after the s? Any video recommendations? Thank you!!!

Answer 1

I think what you wrote is (1/2)∫u³du = (u⁴)/8, where the ∫ is the integral sign.

To see where this comes from, you must first know that the derivative (with respect to u) of u⁴= 4u³

Dividing both sides of the above by 8, you can see that the derivative (again, with respect to u) of (u⁴)/8 = (4u³)/8, which we can simplify to (u³)/2 which can be written as (1/2)u³

Therefore ∫(1/2)u³du = (u⁴)/8

And you can take a constant factor (in this case, the 1/2) out of the integrand and write it in front of the integral sign ∫, so

(1/2) ∫u³du = ∫(1/2)u³du = (u⁴)/8

Therefore

(1/2) ∫u³du = (u⁴)/8 which I think is what you wrote. Thanks.