Evaluate the Integral integral of 4/(x^3+4x) with respect to x

Problem

(∫_^)(4/(x3+4*x)*d(x))

Solution

1. Factor the denominator to prepare for partial fraction decomposition.

x3+4*x=x*(x2+4)

2. Set up the partial fraction decomposition for the integrand.

4/(x*(x2+4))=A/x+(B*x+C)/(x2+4)

3. Solve for the constants A B and C by multiplying by the common denominator.

4=A*(x2+4)+(B*x+C)*x

4. Expand and group the terms by powers of x

4=(A+B)*x2+C*x+4*A

5. Equate the coefficients to find the values of the constants.

4*A=4⇒A=1

C=0

A+B=0⇒B=−1

6. Rewrite the integral using the decomposed fractions.

(∫_^)(4/(x3+4*x)*d(x))=(∫_^)(1/x*d(x))−(∫_^)(x/(x2+4)*d(x))

7. Integrate each term separately, using a usubstitution for the second term where u=x2+4 and d(u)=2*x*d(x)

(∫_^)(1/x*d(x))=ln(x)

(∫_^)(x/(x2+4)*d(x))=1/2*ln(x2+4)

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

ln(x)−1/2*ln(x2+4)+C

Final Answer

(∫_^)(4/(x3+4*x)*d(x))=ln(x)−1/2*ln(x2+4)+C

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