∫ 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

2 Views|Posted 5 months ago

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Answered by

Raj Pandey | Contributor-Level 9

5 months ago

Correct Option - 1

Detailed Solution:

$\int {c o t}^{- 1} ? (\frac{1}{1 + x}) d x$

$= \int {t a n}^{- 1} ? (1 + x) d x$

$= x {t a n}^{- 1} ? (1 + x) - \int \frac{x}{1 + (1 + x)^{2}} d x; = x {t a n}^{- 1} ? (1 + x) - \int \frac{x}{x^{2} + 2 x + 2} d x$

$= x {t a n}^{- 1} ? (1 + x) - \int \frac{\frac{1}{2} (2 x + 2) - 1}{x^{2} + 2 x + 2} d x = x {t a n}^{- 1} ? (1 + x) - \frac{1}{2} l o g ? (x^{2} + 2 x + 2) + \int \frac{d x}{(x + 1)^{2} + 1}$

$= x {t a n}^{- 1} ? (1 + x) - \frac{1}{2} l o g ? (x^{2} + 2 x + 2) + {t a n}^{- 1} ? (x + 1) + c$

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