Evaluate the Integral integral of x/(1+4x^2) with respect to x

Problem

(∫_^)(x/(1+4*x2)*d(x))

Solution

1. Identify the substitution method as the most efficient approach because the numerator x is a multiple of the derivative of the denominator 1+4*x2

2. Define the substitution variable u=1+4*x2

3. Differentiate u with respect to x to find d(u)=8*x*d(x)

4. Rearrange the differential to solve for the terms in the integral: 1/8*d(u)=x*d(x)

5. Substitute the variables into the integral to rewrite it in terms of u

(∫_^)(1/u⋅1/8*d(u))

6. Factor out the constant coefficient from the integral.

1/8*(∫_^)(1/u*d(u))

7. Integrate using the rule (∫_^)(1/u*d(u))=ln(u)+C

1/8*ln(u)+C

8. Back-substitute the original expression for u to get the final result in terms of x

1/8*ln(1+4*x2)+C

Final Answer

(∫_^)(x/(1+4*x2)*d(x))=1/8*ln(1+4*x2)+C

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