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

Problem

(∫_^)(x/(x2+1)*d(x))

Solution

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

2. Define the substitution variable u=x2+1

3. Differentiate u with respect to x to find d(u)=2*x*d(x) which implies x*d(x)=1/2*d(u)

4. Substitute the expressions for u and x*d(x) into the integral.

(∫_^)(1/u⋅1/2*d(u))

5. Factor out the constant 1/2 from the integral.

1/2*(∫_^)(1/u*d(u))

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

1/2*ln(u)+C

7. Back-substitute u=x2+1 to return to the original variable x Since x2+1 is always positive, absolute value bars can be replaced with parentheses.

1/2*ln(x2+1)+C

Final Answer

(∫_^)(x/(x2+1)*d(x))=1/2*ln(x2+1)+C

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