Question

Evaluate the integral

\(I = \int {\frac{{{x^4} + 3{x^2} + 1}}{{{x^5} + 5{x^3} + 5x}}dx} \)

Short answer

Multiply numerator and dominator by 5 to get your answer in simple substitution

Step-by-step solution

Step 1: Substitution step

\(I = \int {\frac{{{x^4} + 3{x^2} + 1}}{{{x^5} + 5{x^3} + 5x}}dx} \)

\( = \frac{1}{5}\int {\frac{{5{x^4} + 15{x^2} + 5}}{{{x^5} + 5{x^3} + 5x}}dx} \)

Substitute\({x^5} + 5{x^3} + 5x = t\)

\(5{x^4} + 15{x^2} + 5dx = dt\)

\(I = \frac{1}{5}\int {\frac{{dt}}{t}} \)

Integration step

\(I = \frac{1}{5}\int {\frac{{dt}}{t}} \)

\( = \frac{1}{5}\log |t| + c\)

\( = \frac{1}{5}\log |{x^5} + 5{x^3} + 5x| + c\)

Therefore, \(I = \int {\frac{{{x^5} + 3{x^2} + 1}}{{{x^5} + 5{x^3} + 5x}}dx} = \frac{1}{5}\log |{x^5} + 5{x^3} + 5x| + c\)