Evaluate ∫ ln (8x) dx.

Step-by-step explanation:

So we have the integral:

First, let's do a substitution to simplify the inside of natural log. Let y equal to 8x. So:

Differentiate with respect to x:

Divide both sides by 8:

Substitute this into our original integral. Therefore:

Move the co-efficient outside:

Now, perform integration by parts. Integration by parts is as follows:

Note that our integral is the same as saying:

Let u equal the natural log and let dv equal 1 dy. Therefore:

Differentiate:

And let dv equal 1 dy. Thus:

Integrate:

Perform integration by parts:

Simplify:

Evaluate the integral:

Substitute 8x for y:

Distribute:

Constant of Integration:

And we have our answer.

Edit: Typo

find the magnitude of the trig term  

csc

(

x

)

−

1

by taking the absolute value of the coefficient.

1

find the lower bound of the range.

tap for more

y

=

−

2

find the upper bound of the range.

tap for more

y

=

0

the range is  

y

≤

−

2

or  

y

≥

0

.

(

−

∞

,

−

2

]

∪

[

0

,

∞

)

{

y

|

y

≤

−

2

,

y

≥

0

}

step-by-step explanation: