Evaluate the Integral

Problem

(∫_^)((x^2)/(x^4−2*x^2−8)*d(x))

Solution

1. Factor the denominator by treating it as a quadratic in terms of x^2

x^4−2*x^2−8=(x^2−4)*(x^2+2)

2. Decompose the integrand into partial fractions using the form (x^2)/((x^2−4)*(x^2+2))=A/(x^2−4)+B/(x^2+2)

x^2=A*(x^2+2)+B*(x^2−4)

3. Solve for the constants A and B by substituting values for x^2

Let *x^2=4⇒4=A(6)⇒A=2/3

Let *x^2=−2⇒−2=B*(−6)⇒B=1/3

4. Rewrite the integral using the partial fraction decomposition.

(∫_^)((x^2)/(x^4−2*x^2−8)*d(x))=(∫_^)((2/3)/(x^2−4)*d(x))+(∫_^)((1/3)/(x^2+2)*d(x))

5. Factor the first denominator further to apply partial fractions again or use the standard integral formula (∫_^)(1/(x^2−a^2)*d(x))=1/(2*a)*ln((x−a)/(x+a))

(∫_^)((2/3)/(x^2−4)*d(x))=2/3⋅1/(2*(2))*ln((x−2)/(x+2))=1/6*ln((x−2)/(x+2))

6. Apply the standard integral formula (∫_^)(1/(x^2+a^2)*d(x))=1/a*arctan(x/a) to the second term.

(∫_^)((1/3)/(x^2+2)*d(x))=1/3⋅1/√(,2)*arctan(x/√(,2))=1/(3√(,2))*arctan(x/√(,2))

7. Combine the results and add the constant of integration C

1/6*ln((x−2)/(x+2))+√(,2)/6*arctan(x/√(,2))+C

Final Answer

(∫_^)((x^2)/(x^4−2*x^2−8)*d(x))=1/6*ln((x−2)/(x+2))+√(,2)/6*arctan(x/√(,2))+C

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