displaystyle ∫ ((x+1)2/x(x2+1)) dX is equal to

Question

displaystyle ∫ ((x+1)2/x(x2+1)) dX is equal to (A) log |x (x2+1)|+C (B) log |x |+C (C) log |x |+2 tan-1 x+C (D) log ((1/1+x2))+C

Tardigrade Admin · Accepted Answer

Let I =∫ ((x+1)2/x(x2+1)) d x =∫ (x2+1+2 x/x(x2+1)) d x =∫ (x2+1/x(x2+1)) d x+∫ (2 x/x(x2+1)) d x =∫ (1/x) d x+2 ∫ (1/x2+1) d x = log |x|+2 tan -1 x+C

Q. $\int \frac{( x + 1 ) ^{2}}{x ( x ^{2} + 1 )}$ dX is equal to

$l o g ∣ ∣ x (x^{2} + 1) ∣ ∣ + C$

$l o g ∣ x ∣ + C$

$l o g ∣ x ∣ + 2 t a n^{- 1} x + C$

$l o g (\frac{1}{1 + x ^{2}}) + C$

$2 l o g ∣ x ∣ + t a n^{- 1} x + C$

Let $I = \int \frac{( x + 1 ) ^{2}}{x ( x ^{2} + 1 )} d x$
$= \int \frac{x ^{2} + 1 + 2 x}{x ( x ^{2} + 1 )} d x$
$= \int \frac{x ^{2} + 1}{x ( x ^{2} + 1 )} d x + \int \frac{2 x}{x ( x ^{2} + 1 )} d x$
$= \int \frac{1}{x} d x + 2 \int \frac{1}{x ^{2} + 1} d x$
$= lo g ∣ x ∣ + 2 tan^{- 1} x + C$

Q. ∫x(x2+1)(x+1)2​ dX is equal to

log∣∣​x(x2+1)∣∣​+C

log∣x∣+C

log∣x∣+2tan−1x+C

log(1+x21​)+C

2log∣x∣+tan−1x+C