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Q.
The value of the integral $\int\limits_1^{16} \frac{ dx }{\sqrt{ x }+\sqrt[4]{ x }}$ is
Integrals
Solution:
Put $x=t^4 \Rightarrow d x=4 t^3 d t$
$I =\int\limits_1^2 \frac{4 t ^3}{ t ^2+ tt }=\int\limits_1^2 \frac{4 t ^2}{1+ t } dt =4\left[\int\limits_1^2 \frac{\left( t ^2-1\right)+1}{1+ t } dt \right]=4\left[\int_1^2( t -1) dt +\int\limits_1^2 \frac{ dt }{1+ t }\right] $
$=4\left[\left(\frac{ t ^2}{2}- t \right)_1^2+\left.\ln (1+ t )\right|_1 ^2\right]=4\left[(0)-\left(-\frac{1}{2}\right)+\ln 3-\ln 2\right]=\left[\frac{1}{2}+\ln \frac{3}{2}\right]=4 \ln \frac{3}{2}+2$