(Solved):Show the long division ? ( 3 x^4 + 3 x^2 + 1 ) x^2 + 1 d x View Answer…



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




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.


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

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


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


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

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