Saved Bookmarks
| 1. |
Evaluate: `int("x"^2+1)/(("x"+1)^2)"dx"` |
|
Answer» `int (x^2+1)/(x+1)^2 dx = int (x^2+1+2x)/(x+1)^2 dx - int (2x)/(x+1)^2 dx` `= int (x+1)^2/(x+1)^2 dx - int (2x)/(x+1)^2 dx` `= int dx - int (2x)/(x+1)^2 dx` `= int dx - int (2(x+1))/(x+1)^2 dx + int 2/(x+1)^2 dx` Let `(x+1)^2 = t => 2(x+1)dx = dt` Then, our integral becomes, `= int dx - int dt/t + int 2/(x+1)^2 dx` `= x -logt -2/(x+1)+c` `= x-log(x+1)^2-2/(x+1)+c` `= x-2log(x+1)-2/(x+1)+c`, which is the required value of given integral. |
|