Question

Help me answer this indefinite integral question please

Answer 1

See below.

Explanation:

(i)
`\int (3)/(x) \, dx`

1) Use Constant Factor Rule:
`\int cf(x) \, dx=c\int f(x)`.

`3\int (1)/(x) \, dx`

2) The derivative of
`ln{x}` is
`(1)/(x)`.

`3\text{ln}x`

3) Add constant.

`3\text{ln}x+C`

------------------------------------------------------------------

(ii)
`\int(1)/(4x) dx`

1) Use Constant Factor Rule:
`\int cf(x) \, dx=c\int f(x)`.

`(1)/(4) \int(1)/(x) dx`

2) The derivative of
`\text{ln}x` is
`(1)/(x)`.

`\frac{\text{ln}x}{4}`

3) Add constant.

`\frac{\text{ln}x}{4} +C`

----------------------------------------------------------------------

(iii)
`\int(1)/(x-5) dx`

1) Use Integration by Substitution.

Let
`u=x-5,du=dx`

2) Using
`u` and
`du` above, rewrite
`\int(1)/(x-5) dx`.

`\int(1)/(u) du`

3) Substitute
`u=x-5` back into the original integral.

`\text{ln}(x-5)`

4) Add constant.

`\text{ln}(x-5)+C`

------------------------------

Thanks.