Help Evaluate?

Question

Evaluate

(from 1 to 0) 6x^5(x^6 + 25)^1/2 dx

Please show step by step work

a) [52(26)^1/2 + 250]/ 3 ~171.72

b) [52(26)^1/2 - 250]/18 ~0.84

c) 2(26)^1/2 -10 ~0.2

d) [52(26)^1/2 -250]/3 ~5.05

Answer 1

OK

The given question is (from 1 to 0) 6x^5(x^6 + 25)^1/2 dx

Let (x^6 + 25) = t
differentiate.
gives, 6x^5 dx = dt

From the above two equationss we can write the given equation by you as follows.
(from 1 to 0 Integral) 6x^5(x^6 + 25)^1./2 dx
gives, (from 1 to 0 Integral) (x^6 + 25)^1/2 * 6x^5 dx
now substitute the assumed equations. Ans also the limita changes as,
when x = 1, t = 1^6 + 25 = 26
when x = 0, t = 0 + 25 = 25

gives, (from 26 to 25 integral) t^1/2 dt

I think you know that integral y^n dy = [y^(n+1)] / n+1

So it gives 2/3 t^3/2 (from 26 to 25 )
substituting upper and lower limit and subtrating.

= 2/3 { 25^3/2 - 26^3/2 }
= 2/3 { 125 - 132.574 }
= 2/3 { -7.574 }
= -5.049
= -5.05

It is the fourth option but with negative sign.

Hop you had understood this problem simply.

Answer 2

I don't know really. I hope someone helps you out.