Verified answer

Let u = 21x^4 + 42x^2 + 1

Then du = 84x^3 + 84x dx = 84 * (x^3 + x) dx

So now you have: (1/84) * sqrt(u) du = (1/84 ) * u^(3/2) / (3/2) = (2 / (84 * 3)) * u^(3/2) =

(1 /126) * ((21x^4) + 42x^2 + 1)^(3/2)

At 0 this is (1/126)

At 1 this is (1/126) * (21 + 42 + 1)^(3/2) =

(1/126) * 64^(3/2) = (1/126) * 8^3 = (512/126)

Not forgetting a, we have (511/126) <----- Answer as a fraction

= 4.05556 as a decimal <------------- Answer as a decimal

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Use u-substitution on u = 21x^4 + 42x^2 + 1, so that du = 84x^3 + 84x dx, which means dx = du / 84(x^3 + x). This leaves you with the integral of (1/84) u^(1/2) du.