Evaluate the Integral

Problem

(∫_^)((4*x)/(x2+9)*d(x))

Solution

1. Identify the structure of the integrand and notice that the numerator 4*x is a multiple of the derivative of the denominator x2+9

2. Apply substitution by letting u=x2+9

3. Calculate the differential d(u) by differentiating u with respect to x

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

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

4. Rewrite the integral in terms of u by substituting 2*x*d(x)=d(u) and x2+9=u Since the original numerator is 4*x we use 4*x*d(x)=2*(2*x*d(x))=2*d(u)

(∫_^)((4*x)/(x2+9)*d(x))=(∫_^)(2/u*d(u))

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

2*(∫_^)(1/u*d(u))=2*ln(u)+C

6. Back-substitute the original expression for u to get the final result in terms of x Since x2+9 is always positive, the absolute value bars can be replaced with parentheses.

2*ln(x2+9)+C

Final Answer

(∫_^)((4*x)/(x2+9)*d(x))=2*ln(x2+9)+C

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