Question

Show the long division {eq}int frac{(3x^{4}+3x^{2}+1)}{x^{2}+1}dx

{/eq}

EXPERT ANSWER

The task is to evaluate the integration of $$displaystyle I = int frac{(3x^{4}+3x^{2}+1)}{x^{2}+1}dx

$$

Apply the long division.

Step 1. Divide the leading coefficients of the numerator {eq}3x^4+3x^2+1

{/eq} and the divisor {eq}x^2+1.

{/eq}

Then we get Quotient {eq}=3x^2

{/eq} and Remainder {eq}=1.

{/eq}

Therefore, {eq}3x^2+frac{1}{x^2+1}.

{/eq}

Then integration can be written as:

$$displaystyle = int 3x^2+frac{1}{x^2+1}dx

$$

Apply the sum rule.

$$displaystyle = int 3x^2dx+int frac{1}{x^2+1}dx

$$

Take the constant out.

$$displaystyle = 3cdot int x^2dx+int frac{1}{x^2+1}dx

$$

Use the power rule and