Evaluate the integral of 1 / (x * (x4+1)) dx.
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Step-by-Step Solution
Step 1: Rewrite the integrand
To simplify the integral, we multiply the numerator and denominator by x3. This allows us to make a substitution involving x4 later.
Step 2: Substitute u=x4
We introduce a substitution to simplify the integral further. Let u=x4. Then, we find the differential du by differentiating u with respect to x. This allows us to replace x3dx with 41du.
Step 3: Substitute into the integral
Now we substitute u and du into the integral. The integral becomes 41∫u(u+1)1du.
Step 4: Use partial fraction decomposition
We use partial fraction decomposition to break down the integrand u(u+1)1 .Wesetu(u+1)1=uA+u+1B. By solving for A and B ,wefindA=1andB=−1.
Step 5: Integrate with respect to u
Now we integrate the decomposed fractions. The integral of u1 is ln∣u∣ and the integral of u+11isln∣u+1∣. We combine the logarithms using the property lna - lnb=ln(ba).
Step 6: Substitute back x4 for u
Finally, we substitute x4 back in for u to express the result in terms of x$.