Evaluate the following: dt 3t-2t^2 - Vidyadip

Question

Evaluate the following:
$\int\frac{\text{dt}}{\sqrt{3\text{t}-2\text{t}^2}}$

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Answer

Let $\text{I}=\int\frac{\text{dt}}{\sqrt{3\text{t}-2\text{t}^2}}$
$=\frac{1}{2}\int\frac{\text{dt}}{\sqrt{-\Big(\text{t}^2-\frac{3}{2}\text{t}\Big)}}$
$=\frac{1}{2}\int\frac{\text{dt}}{\sqrt{\Big(\frac{3}{4}\Big)^2-\Big(\text{t}-\frac{3}{4}\Big)^2}}$
$=\frac{1}{\sqrt{2}}\sin^{-1}\bigg(\frac{\text{t}-\frac{3}{4}}{\frac{3}{4}}\bigg)+\text{C}$ $=\frac{1}{\sqrt{2}}\sin^{-1}\Big(\frac{4\text{t}-3}{3}\Big)+\text{C}$

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