MHTCET 20 Apr 2026 Shift 2 Paper

Question : 130 of 150

Marks: +1, -0

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

Solution:

Concept:

The integrand is simplified by completing the square, then the standard inverse sine integral formula is applied.

Explanation:

Rewrite

2x - x^2

by completing the square.

2x - x^2 = -(x^2 - 2x)

Since

x^2 - 2x = (x-1)^2 - 1

, we get:

2x - x^2 = 1 - (x-1)^2

Therefore, the integral becomes:

\int \frac{1}{\sqrt{1-(x-1)^2}} \, dx

Use the standard formula:

\int \frac{du}{\sqrt{1-u^2}} = \sin^{-1} u + C

Here, let

u = x-1

, so

du = dx

Thus, the integral evaluates to:

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

Answer:

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

Hence, the correct option is Option A.

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