Evaluate the integral : ∫ (ax^3+bx)/(x^4+c^2)dx

Question

Evaluate the integral :

\(\int\frac{ax^3+bx}{x^4+c^2}\)dx

Answer 1

I =  \(\int\frac{ax^3+bx}{x^4+c^2}\)dx

As we can see that there is a term of x³ in numerator and derivative of x⁴ is also 4x³. 

So there is a chance that we can make substitution for x⁴ + c² and I can be reduced to a fundamental integration but there is also a x term present. 

So it is better to break this integration.

To make the substitution, 

I₁ can be rewritten as,

∴ Let, 

x⁴ + c² = u 

⇒ du = 4x³ dx 

I₁ is reduced to simple integration after substituting u and du as :

∵ We have derivative of x² in numerator and term of x² in denominator. 

So we can apply method of substitution here also.

As denominator doesn’t have any square root term. 

So one of the following two integrals will solve the problem.