Evaluate the Integral

Problem

(∫_^)((4*x2)/(x2+9)*d(x))

Solution

1. Perform polynomial division to rewrite the integrand, since the degree of the numerator is equal to the degree of the denominator.

(4*x2)/(x2+9)=(4*(x2+9)−36)/(x2+9)

2. Split the fraction into two separate terms.

(4*(x2+9))/(x2+9)−36/(x2+9)=4−36/(x2+9)

3. Rewrite the integral using the linearity property of integration.

(∫_^)((4−36/(x2+9))*d(x))=(∫_^)(4*d(x))−36*(∫_^)(1/(x2+3)*d(x))

4. Apply the integration rules for a constant and for the inverse tangent form (∫_^)(1/(x2+a2)*d(x))=1/a*arctan(x/a)+C

(∫_^)(4*d(x))=4*x

36*(∫_^)(1/(x2+3)*d(x))=36*(1/3*arctan(x/3))

5. Simplify the expression and add the constant of integration C

4*x−12*arctan(x/3)+C

Final Answer

(∫_^)((4*x2)/(x2+9)*d(x))=4*x−12*arctan(x/3)+C

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