Suppose that α and β are two continuous and differentiable functions. Suppose also that $α′(x) =β(x)$, $β^3(x)= (α(x) − 1)^2$, $α(a) = 0$,and $α(b) = 2$. Using this information, compute the definite integral $\int_a^b α^2(t)β^4(t) dt.$
This is how I did it.
$\int_a^b α^2(t)β^3(t) β(t) dt$
Substitute $(α(t) − 1)^2$ for $β^3(t)$ to get,
$\int_a^b α^2(t)(α(t) − 1)^2β(t) dt$
Using u-substitution, I set u=$α(t)$, which gave me du=$α′(t)$dt and I substituted $α′(t)$ for $β(x)$ to get du=$β(x)$dt
Substituting for the variables,$\int_{u(a)=0}^{u(b)=2} u^2(u − 1)^2du$
Then I arrived at, $\frac{u^5}{5}-\frac{u^4}{2}+\frac{u^3}{3}|_0^2=\frac{16}{15}$
Am I completely wrong?