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

Problem

(∫_^)(x/(x2+4)*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+4

2. Define the substitution variable u=x2+4

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 variables into the integral to rewrite it in terms of u

(∫_^)(x/(x2+4)*d(x))=(∫_^)(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 the original expression x2+4 for u Since x2+4 is always positive, the absolute value bars can be replaced with parentheses.

1/2*ln(x2+4)+C

Final Answer

(∫_^)(x/(x2+4)*d(x))=1/2*ln(x2+4)+C

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