Question

Integrate {eq}displaystyle frac{(x^{5} + x – 1)}{(x^{3} + 1)}dx{/eq}.

EXPERT ANSWER

The task is to evaluate the integration of $$displaystyle I = frac{(x^{5} + x – 1)}{(x^{3} + 1)}dx

$$

Apply the long division.

Step 1. Divide the leading coefficients of the numerator {eq}x^5+x-1

{/eq} and the divisor {eq}x^3+1.

{/eq}

Then we get Quotient {eq}=x^2

{/eq} and Remainder {eq}= -x^2+x-1.

{/eq}

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

{/eq}

So, we get integration in the form as

$$begin{align*}

displaystyle &=int x^2+frac{-x^2+x-1}{x^3+1} dx\

displaystyle &= int x^2-frac{x^2-x+1}{(x+1)(x^2-x+1)} dx

end{align*}

$$

Apply the sum rule.

$$displaystyle =