Find out the answer to "Evaluate the following integrals. ∫dx(x−1)(3−x)"

Tabansi

Answered question

2020-10-23

Evaluate the following integrals.

\int \frac{d x}{\sqrt{(x - 1) (3 - x)}}

Neelam Wainwright

Skilled2020-10-24Added 102 answers

Step 1: Given that

\int \frac{d x}{\sqrt{(x - 1) (3 - x)}}

Step 2: Formula Used

\int \frac{1}{\sqrt{a^{2} - x^{2}}} d x = \sin^{- 1} (\frac{x}{a}) + C

Step 3: Solve
We have,

\int \frac{d x}{\sqrt{(x - 1) (3 - x)}} = \int \frac{d x}{\sqrt{3 x - x^{2} - 3 + x}}

= \int \frac{d x}{\sqrt{- x^{2} + 4 x - 3}}

= \int \frac{d x}{\sqrt{- (x^{2} - 4 x + 3)}}

= \int \frac{d x}{\sqrt{- (x^{2} - 4 x + 2^{2} - 2^{2} + 3)}}

= \int \frac{d x}{\sqrt{- ({(x - 2)}^{2} - 4 + 3)}}

= \int \frac{d x}{\sqrt{- ({(x - 2)}^{2} - 1)}}

= \int \frac{d x}{\sqrt{1 - {(x - 2)}^{2}}}

= \sin^{- 1} (x - 2) + C

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