Integrate the function: a x+b - Vidyadip

Question

Integrate the function: $\sqrt{a x+b}$

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Answer

Let ax + b = t
$\Rightarrow$ adx = dt
$\Rightarrow$ $d x=\frac{1}{a} d t$
$\Rightarrow \int(a x+b)^{\frac{1}{2}}=\frac{1}{a} \int t^{\frac{1}{2}} d t$
= $\frac{1}{a}\left(\frac{t^{\frac{3}{2}}}{\frac{3}{2}}\right)+C$
$\Rightarrow\frac{2}{3 a}(a x+b)^{\frac{3}{2}}+C$

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