How do you integrate a radical function such as f(x) = 7squareroot 4x5 - 36square root x5 - 114 square root x + 33 square root x? describe your startegy.

Question

Calculus Question

Answer 1

If your not getting an answer, then there is probably some error when you rewrite the f(x) using fraction exponents.  Let's go over this part.

 

f(x) = 7√(4x⁵) - 36√(x⁵) - 114√(x) + 33√(x)

 

 

Breaking up the radicals by their factors.  We can also combine like radicals.

 

f(x) = 7 √4 √x² √x² √x  -  36 √x² √x² √x  -  114 √x  +  33 √x

 

f(x) = 14x² √x  -  36x² √x  -  81√x

 

f(x) = -22x² √x  -  81x^1/2

 

 

Turn the radicals into fractional exponents.

 

f(x) = -22 x² x^1/2  -  81 x² x^1/2

 

f(x) = -22x^5/2 - 81x^1/2

 

 

Okay, now you can use the integration method that Muhammad suggested to easily integrate f(x).  Integrate term by term.

 

 

Keep in mind that when you integrate terms that have constants, the constant can go in front of the integral sign.  For example:

 

∫5x² dx = 5 ∫x² dx

 

 

So we only integrate what is inside the radical, and we can ignore the constant.

 

 

 

Answer 2

First....you must take all the expressions under every radical & rewrite them as expressions being raised to fractional exponents. Then you integrate each individual expression within the function via the power rule of integration.

 

∫xⁿ = xⁿ⁺¹/(n+1) + C