Estimation of a defined integral I need to show that $$\left|\int\limits_{0}^{\frac{1}{2}}\frac{(2t-1)^3}{(\sqrt{1+t})^7}dt\right|<\frac{16}{125}$$ Evaluating it would be my last hope, but it wouldn't be easy wither. Is there a trick for that kind of problems?

The denominator is greater than 1. So the value is less than $\int_0^{0.5}(1-2t)^3dt=\frac{1}{8}=\frac{16}{128}<\frac{16}{125}$.