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∫ cot?¹(1 - x + x² - x³ + x? - x? + ...) dx (|x|<1)

Option 1 -

(x+1)tan⁻¹(x+1) - log√(1 + (x+1)²) + C

Option 2 -

(x-1)tan⁻¹(x+1) - log√(1 - (x+1)²) + C

Option 3 -

(x+1)tan⁻¹(x-1) + log√(1 + (x+1)²) + C

Option 4 -

(x-1)tan⁻¹(x-1) - log√(1 - (x-1)²) + C

0 2 Views | Posted a month ago
Asked by Shiksha User

  • 1 Answer

  • R

    Answered by

    Raj Pandey | Contributor-Level 9

    a month ago
    Correct Option - 1


    Detailed Solution:

    ∫ c o t - 1 ? 1 1 + x d x

    = ∫ t a n - 1 ? ( 1 + x ) d x

    = x t a n - 1 ? ( 1 + x ) - ∫ x 1 + ( 1 + x ) 2 d x ; = x t a n - 1 ? ( 1 + x ) - ∫ x x 2 + 2 x + 2 d x

    = x t a n - 1 ? ( 1 + x ) - ∫ 1 2 ( 2 x + 2 ) - 1 x 2 + 2 x + 2 d x = x t a n - 1 ? ( 1 + x ) - 1 2 l o g ? x 2 + 2 x + 2 + ∫ d x ( x + 1 ) 2 + 1

    = x t a n - 1 ? ( 1 + x ) - 1 2 l o g ? x 2 + 2 x + 2 + t a n - 1 ? ( x + 1 ) + c

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If ∫ − a a ( | x | + | x − 2 | ) d x = 2 2 , ( a > 2 )     a n d     [ x ] denotes the greatest integer ≤ x , then ∫ a − a ( x + [ x ] ) d x  is equal to__________.
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The value of the integral

∫ s i n θ s i n θ ( s i n 6 θ + s i n 4 θ + s i n 2 θ ) 2 s i n 4 θ + 3 s i n 2 θ + 6 1 − c o s 2 θ d θ (where c is a constant of integration)
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