Evaluate the integrals given

Question

∫₀¹ (x⁵-5^x+5x)dx

 

 

∫₁⁴  √y-y/y²  dy                        

Answer 1

I will do the first one for you.

 

Find the indefinite integral first.  We can easily integrate the 1st and last terms using the power rule.

 

(1/6)x⁶ - ∫5^x dx + (5/2)x²

 

 

We need to find the integral of the middle term using some other techniques.

 

Let       u = 5^x 

 

         x = (1 / ln5)(lnu)

 

      dx = du / uln(5)

 

 

Substituting,

 

∫udu / (uln5) =

 

∫1 / (ln5) du =

 

(1 / ln(5)) u =

 

5^x / ln(5)

 

 

Lets check this integral to see if we get back the sub-integrand.

 

(5^x ln5) (ln5) / (ln5)²

 

 

It checks.  Therefore,

 

∫5^x dx = 5^x / ln(5)

 

 

So your definite integral is

 

[(1/6)(1)⁶ - (5 / ln5) + (5/2)(1)²] - [(1/6)(0)⁶ - (1 / ln5)] =

 

(1/6) - (5 / ln5) + (5/2) + (1 / ln5) =

 

-(4 / ln5) + (16 / 6) =

 

-(4 / ln5) + (8 / 3)

 

 

As for the second one, you should rewrite the  radical as rational exponents.  Then apply your integration techniques.  SInce you did not use parenthesis to make it clear, I cannot answer this question for you.