Q. `int` `log(x+1) dx` =? A) `xlogx - x + log(x+1) + c` B) `(x+1)log(x+1) - x +c` C) `log(x/(x+1)) + c` D) `log [(1+x)/(2+x)] + c`

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This can be done using integral by parts.


`u = log(x+1)`

`du = 1/(x+1)dx`

`v = x`

`dv = dx`

`intudv = uv-intvdu`

`intlog(x+1)dx = xlog(x+1)-intx/(x+1)dx`

`intlog(x+1)dx = xlog(x+1)-int(x+1-1)/(x+1)dx`

`intlog(x+1)dx = xlog(x+1)-[int(x+1)/(x+1)dx-int1/(x+1)dx]`

`intlog(x+1)dx = xlog(x+1)-intdx+int1/(x+1)dx`

`intlog(x+1)dx = xlog(x+1)-x+log(x+1)+C` where C is a constant.

`intlog(x+1)dx = (x+1)log(x+1)-x+C`

So the answer is;`intlog(x+1)dx = (x+1)log(x+1)-x+C`

Correct answer is B.


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