∫ x 51( tan -1 x + cot -1 x ) d x

Question

∫ x 51( tan -1 x + cot -1 x ) d x (A) (x52/52)( tan -1 x+ cot -1 x)+c (B) (x52/52)( tan -1 x- cot -1 x)+c (C) (π x52/104)+(π/2)+c (D) (x52/52)+(π/2)+c

Tardigrade Admin · Accepted Answer

∫ x51( tan -1 x+ cot -1 x) d x =∫ x 51 ⋅ (π/2) dx ∵ tan -1 x + cot -1 x =(π/2) =(π x 52/104)+ c =( x 52/52)( tan -1 x + cot -1 x )+ c .

Q. $\int x^{51} (tan^{- 1} x + cot^{- 1} x) d x$

$\frac{x ^{52}}{52} (tan^{- 1} x + cot^{- 1} x) + c$

$\frac{x ^{52}}{52} (tan^{- 1} x - cot^{- 1} x) + c$

$\frac{π x ^{52}}{104} + \frac{π}{2} + c$

$\frac{x ^{52}}{52} + \frac{π}{2} + c$

$\int x^{51} (tan^{- 1} x + cot^{- 1} x) d x$
$= \int x^{51} \cdot \frac{π}{2} d x {∵ tan^{- 1} x + cot^{- 1} x = \frac{π}{2}}$
$= \frac{π x ^{52}}{104} + c = \frac{x ^{52}}{52} (tan^{- 1} x + cot^{- 1} x) + c .$

Q. ∫x51(tan−1x+cot−1x)dx

52x52​(tan−1x+cot−1x)+c

52x52​(tan−1x−cot−1x)+c

104πx52​+2π​+c

52x52​+2π​+c