Check the solution to "Evaluate the integral without using tables. ∫01dx1−x2"

BenoguigoliB

Answered question

2021-11-08

Evaluate the integral without using tables.

\int_{0}^{1} \frac{d x}{\sqrt{1 - x^{2}}}

yagombyeR

Skilled2021-11-09Added 92 answers

Step 1
Given,

I = \int_{0}^{1} \frac{d x}{\sqrt{1 - x^{2}}}

Step 2
Formula used:

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

\int x^{n} d x = \frac{x^{n + 1}}{n + 1} + C

Step 3
Apply the above formula, we get

I = \int_{0}^{1} \frac{d x}{\sqrt{1 - x^{2}}}

= {[\sin^{- 1} (x)]}_{0}^{1}

= \sin^{- 1} (1) - \sin^{- 1} (0)

= \sin^{- 1} (\sin (\frac{π}{2})) - \sin^{- 1} (\sin (0))

= \frac{π}{2} - 0

= \frac{π}{2}

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