Evaluate the Integral integral of x/(1+x^4) with respect to x

Problem

(∫_^)(x/(1+x4)*d(x))

Solution

1. Rewrite the denominator to prepare for a substitution by expressing x4 as a square.

(∫_^)(x/(1+(x2)2)*d(x))

2. Identify a substitution where the derivative of the inner function matches the numerator. Let u=x2

d(u)=2*x*d(x)

3. Adjust the differential to solve for x*d(x)

1/2*d(u)=x*d(x)

4. Substitute the variables into the integral.

(∫_^)((1/2)/(1+u2)*d(u))

5. Factor out the constant from the integral.

1/2*(∫_^)(1/(1+u2)*d(u))

6. Integrate using the standard arctangent rule (∫_^)(1/(1+u2)*d(u))=arctan(u)+C

1/2*arctan(u)+C

7. Back-substitute x2 for u to get the final result in terms of x

1/2*arctan(x2)+C

Final Answer

(∫_^)(x/(1+x4)*d(x))=1/2*arctan(x2)+C

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