Simplify the integrand: Step 1: Simplify the integrand.We start by simplifying the integrand (x4+1)/(x2−1). We can rewrite x4+1 as (x2+1)(x2−1)+2.So, (x4+1)/(x2−1)=(x2+1)+2/(x2−1).
Split the integral: Step 2: Split the integral.Now, integrate each part separately:∫(x2−1x4+1)dx=∫(x2+1)dx+∫x2−12dx.
Integrate the first part: Step 3: Integrate the first part.∫(x2+1)dx=31x3+x+C.
Integrate the second part: Step 4: Integrate the second part.∫x2−12dx=2∫x2−11dx.We use partial fractions for x2−11, which decomposes to 21(x−11−x+11).So, 2∫x2−11dx=2×[21×(ln∣x−1∣−ln∣x+1∣)]=ln∣x−1∣−ln∣x+1∣+C.
Combine the results: Step 5: Combine the results.The integral of (x4+1)/(x2−1) is:(1/3)x3+x+ln∣x−1∣−ln∣x+1∣+C.
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