`int (csc^2(t))/cot(t) dt` Find the indefinite integral.

Expert Answers

An illustration of the letter 'A' in a speech bubbles

`int (csc^2(t))/(cot(t))dt=`

We will use the following formula: 

 `int (f'(x))/(f(x))dx=ln|f(x)|+C.`

We will use the following formula:  

The formula tells us that if we have integral of rational function where numerator is equal to derivative of denominator, then the integral is equal to natural logarithm of the denominator plus some constant. The proof of the formula can be obtained by simply integrating the right-hand side.

Since `(cot(t))'=-csc^2(t)` we first have to modify our integral in order to apply the formula. We will put the minus sign in the numerator but then we must also put minus sign in front of the integral (`-1 times-1=1`).

`-int(-csc^2(t))/cot(t)dt=`

Now we apply the formula.

`-ln(cot(t))+C=`

We can further simplify the result by using rule for logarithm of power: `r log_b x=log_b x^r`

`ln(cot^(-1)(t))+C=`

And since `cot^(-1)(t)=tan(t)` we have

`ln(tan(t))+C` 

Approved by eNotes Editorial Team

We’ll help your grades soar

Start your 48-hour free trial and unlock all the summaries, Q&A, and analyses you need to get better grades now.

  • 30,000+ book summaries
  • 20% study tools discount
  • Ad-free content
  • PDF downloads
  • 300,000+ answers
  • 5-star customer support
Start your 48-Hour Free Trial