`int (x^2 + 2x + 3)/(x^3 + 3x^2 + 9x) dx` Find the indefinite integral.

Expert Answers

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

 `int (x^2+2x+3)/(x^3+3x^2+9x)dx=`

We will use the following formula: `int (f'(x))/(f(x))dx=ln|f(x)|+C`   

The formula tells us that if we have integral of rational function where the numerator is equal to the derivative of the 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 `(x^3+3x^2+9x)'=3x^2+6x+9=3(x^2+2x+3)`  we will first have to slightly modify the integral in order to apply the formula. We will both multiply and divide the integral by 3.

`1/3int (3x^2+6x+9)/(x^3+3x^2+9x)dx=`

Now we apply the formula to obtain the final result.



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