∫ x ⋅ ( ln ( x +√1+ x 2)/√1+ x 2) dx equals :

Question

∫ x ⋅ ( ln ( x +√1+ x 2)/√1+ x 2) dx equals : (A) √1+ x 2 ln ( x +√1+ x 2)- x + c (B) (x/2) ⋅ ln 2(x+√1+x2)-(x/√1+x2)+c (C) ( x /2) ⋅ ln 2( x +√1+ x 2)+( x /√1+ x 2)+ c (D) √1+ x 2 ln ( x +√1+ x 2)+ x + c

Tardigrade Admin · Accepted Answer

use I.B.P. taking ln (x+√1+x2) as the first function and (x/√1+x2) as the second function

Q. $\int x \cdot \frac{l n ( x + 1 + x ^{2} )}{1 + x ^{2}} d x$ equals :

$1 + x^{2} ln (x + 1 + x^{2}) - x + c$

$\frac{x}{2} \cdot ln^{2} (x + 1 + x^{2}) - \frac{x}{1 + x ^{2}} + c$

$\frac{x}{2} \cdot ln^{2} (x + 1 + x^{2}) + \frac{x}{1 + x ^{2}} + c$

$1 + x^{2} ln (x + 1 + x^{2}) + x + c$

use I.B.P. taking $ln (x + 1 + x^{2})$ as the first function and $\frac{x}{1 + x ^{2}}$ as the second function

Q. ∫x⋅1+x2​ln(x+1+x2​)​dx equals :

1+x2​ln(x+1+x2​)−x+c

2x​⋅ln2(x+1+x2​)−1+x2​x​+c

2x​⋅ln2(x+1+x2​)+1+x2​x​+c

1+x2​ln(x+1+x2​)+x+c