indefinite integral??

Question

Find the indefinite integral below.

 

∫(x^(3/4)+x^(5/2) )dx

Answer 1

The integral is the same as anti-derivative.  Think of it as going backwards.

 

Use the rule      ∫Cxⁿ dx = C∫xⁿ = Cx^{(n + 1)} / (n + 1)

 

 

[x^7/4 / (7/4)] + [x^7/2 / (7/2)] + C =

 

(4 / 7)x^7/4 + (2 / 7)x^7/2 + C

 

 

We can put this answer in radical form.

 

[(4 / 7) * ⁴√x⁴ * ⁴√x³] + [(2 / 7) * √x⁶ * √x] + C =

 

[(4 / 7)x * ⁴√x³] + [(2 / 7)x³ √x] + C

 

 

Note that C is an arbitrary constant.