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99 Percentile Qs Bank for JEE MainMathematicsIndefinite Integration

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

Options

  1. A1 2 [ x- cosec x- | ( x 2 ) ( 4 + x 2 ) | ]+c
  2. Bx+ cosec x+ | ( x 2 ) ( 4 + x 2 ) |+c
  3. C1 2 [ x- cosec x- | ( 4 + x 2 ) ( x 2 ) | ]+c
  4. Dx+ cosec x- | ( x 2 ) |+c

Correct answer

C. 1 2 [ x- cosec x- | ( 4 + x 2 ) ( x 2 ) | ]+c

Step-by-step solution

aligned & Take I = x ^2 x- x x+ x (1- 2 x) d x & I = x ^2 x- x x+ x 1-1+2 ^2 x d x & I = 1 2 [ ^2 x x d x- x d x+ x ^2 x d x ] & Let x=t & x d x=d t & I = 1 2 [ 1+ ^2 x x d x- | x+ x|+ d t t^2 ] & I = 1 2 [ cosec x d x+ x x+d x- | x+ x|- 1 t ]+c & I = 1 2 [ | cosec x- x|+ x- cosec x- | x+ x|] & I= 1 2 [ x- cosec x+ | 2 2 x / 2 2 ^ x / 2 ^ x / 2 |- | ^ x / 2 + ^ x / 2 ^ x / 2 - ^ x / 2 | ] & I = 1 2 [ x- cosec x+ | x 2 |- | ( 4 + x 2 ) | ] & I = 1 2 [ x- cosec x- | ( 4 + x 2 ) x 2 | ]+c . & aligned So, option (c) is

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