Evaluate the Integral

Problem

(∫_^)((x2)/(4+x2)*d(x))

Solution

1. Perform polynomial long division or use the algebraic manipulation of adding and subtracting 4 in the numerator to simplify the integrand.

(x2)/(4+x2)=(x2+4−4)/(4+x2)

2. Split the fraction into two separate terms.

(x2+4−4)/(4+x2)=(x2+4)/(x2+4)−4/(x2+4)

3. Simplify the expression before integrating.

1−4/(x2+4)

4. Apply the linearity of integration to integrate each term separately.

(∫_^)(1*d(x))−(∫_^)(4/(x2+4)*d(x))

5. Evaluate the first integral using the power rule.

(∫_^)(1*d(x))=x

6. Evaluate the second integral using the standard integral formula (∫_^)(1/(x2+a2)*d(x))=1/a*arctan(x/a)+C where a=2

(∫_^)(4/(x2+4)*d(x))=4⋅1/2*arctan(x/2)

7. Simplify the coefficient and combine the results, adding the constant of integration C

x−2*arctan(x/2)+C

Final Answer

(∫_^)((x2)/(4+x2)*d(x))=x−2*arctan(x/2)+C

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