Question

Determine the following integrals by reversing differentiation.

Answer 1

Show that\[\frac{ d }{ dx }(x+1)^2=2x+2\]

Answer 2

Hence,find\[\int\limits_{}^{}(x+1)dx\]

Answer 3

so... where are you stuck at?

Answer 4

The second one

Answer 5

\(\Large \dfrac{d}{dx}f(x) = g(x) \implies \int g(x) dx = f(x)+c \)

Answer 6

\(\Large \dfrac{d}{dx}(x+1)^2 = 2x+2 \implies \int (2x+2)dx = (x+1)^2 +c\)

good so far?

Answer 7

yes

Answer 8

factor out 2 from 2x+2 and shift that to right side

Answer 9

\[\int\limits_{}^{}(x+1)dx=\frac{ 1 }{ 2 }(x+1)^2\]

Answer 10

that +c
its the constant of integration

Answer 11

but the answer say it's\int\limits_{}^{}(x+1)dx=\frac{ 1 }{ 2 }(x+1)^2
that's why i'm confused whether need to put +c or not

Answer 12

wait i don't think i need to put +c

Answer 13

you know why do we put the +c there ?

Answer 14

no ;/

Answer 15

what will be d/dx [(x+1)^2 + 4] ?
or
d/dx [(x+1)^2 + 7] ?
or
d/dx [(x+1)^2 + 123.456] ?

Answer 16

same? 2x+2 right?

Answer 17

yes

Answer 18

so in general,
d/dx (x+1)^2 + c is 2x+2

and hence while reversing it, we usually take the general case.
integral, 2x+2 = (x+1)^2 +c.

In this case, since we are using (x+1)^2 specifically, so we will not be considering the general case.
hence the answer says its just 1/2 (x+1)^2
makes sense?

Answer 19

oh i c...yes,it makes sense now :)

Answer 20

Thank you @hartnn :)

Answer 21

welcome ^_^