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Solve the following.
Evaluate: ∫ (dx)/(√(5x - 4x²)).
Evaluate: $\int \frac{dx}{\sqrt{5x - 4x^2}}$.
Answer
Answer
Integral: $\int \left(1/sqrt(5x - 4x^2)\right) dx$
$5x - 4x^2 = 4\left(\frac{5x}{4} - x^2\right) = 4\left[\left(\frac{5}{8}\right)^2 - \left(x - \frac{5}{8}\right)^2\right]$
$\int\frac{dx}{\sqrt{5x - 4x^2}} = \frac{1}{2}\int\frac{dx}{\sqrt{\left(\frac{5}{8}\right)^2 - \left(x - \frac{5}{8}\right)^2}}$
Using $\int\frac{dx}{\sqrt{a^2 - u^2}} = \sin^{-1}\frac{u}{a}$:
Hence $\int\frac{dx}{\sqrt{5x - 4x^2}} = \frac{1}{2}\sin^{-1}\left(\frac{8x - 5}{5}\right) + C$.
Final answer: $\frac{1}{2}\sin^{-1}\left(\frac{8x - 5}{5}\right) + C$