QJEEM21Mac320: JEE Main 2021 Mathematics - Class 12 JEEM21Mac320 Solution: The integral ∫ ( 2 x − 1 ) cos ⁡ ( 2 x − 1 ) 2 + 5 4 x 2 − 4 x…

Question

Verified Solution • 2025–26 Session moderate

The integral

\int \frac{(2 x - 1) \cos \sqrt{{(2 x - 1)}^{2} + 5}}{\sqrt{4 x^{2} - 4 x + 6}} d x

is equal to (where c is a constant of integration)

Accepted Answer

\int \frac{(2 x - 1) \cos \sqrt{{(2 x - 1)}^{2} + 5}}{\sqrt{{(2 x - 1)}^{2} + 5}} d x

(2 x - 1)^{2} + 5 = t^{2}

2 (2 x - 1) 2 d x = 2 t d t

2 \sqrt{t^{2} - 5} d x = t d t

So,

\int \frac{\sqrt{t^{2} - 5} \cos t}{2 \sqrt{t^{2} - 5}} d t = \frac{1}{2} \sin t + c

= \frac{1}{2} \sin \sqrt{{(2 x - 1)}^{2} + 5} + c

Step-by-Step Verified Solution