Evaluate the integral. (Remember to use absolute values where appropriate. Use \( C \) for the constant of integration.) \[ \int \frac{5}{(x-1)\left(x^{2}+4\right)} d x \] Need Help? Watch li Master II

Calculate the integral \( \int \frac{5}{(x-1)(x^2+4)} dx \). Evaluate the integral by following steps: - step0: Evaluate using substitution: \(\int \frac{5}{\left(x-1\right)\left(x^{2}+4\right)} dx\) - step1: Multiply the terms: \(\int \frac{5}{x^{3}+4x-x^{2}-4} dx\) - step2: Rewrite the expression: \(\int 5\times \frac{1}{x^{3}+4x-x^{2}-4} dx\) - step3: Use properties of integrals: \(5\times \int \frac{1}{x^{3}+4x-x^{2}-4} dx\) - step4: Rewrite the fraction: \(5\times \int \left(\frac{1}{5x-5}-\frac{1+x}{5x^{2}+20}\right) dx\) - step5: Use properties of integrals: \(5\left(\int \frac{1}{5x-5} dx+\int -\frac{1+x}{5x^{2}+20} dx\right)\) - step6: Calculate: \(5\times \int \frac{1}{5x-5} dx+5\times \int -\frac{1+x}{5x^{2}+20} dx\) - step7: Evaluate the integral: \(\ln{\left(\left|x-1\right|\right)}+5\times \int -\frac{1+x}{5x^{2}+20} dx\) - step8: Evaluate the integral: \(\ln{\left(\left|x-1\right|\right)}-\frac{1}{2}\arctan\left(\frac{x}{2}\right)-\frac{1}{2}\ln{\left(x^{2}+4\right)}\) - step9: Subtract the terms: \(\ln{\left(\frac{\left|x-1\right|}{\left(x^{2}+4\right)^{\frac{1}{2}}}\right)}-\frac{1}{2}\arctan\left(\frac{x}{2}\right)\) - step10: Add the constant of integral C: \(\ln{\left(\frac{\left|x-1\right|}{\left(x^{2}+4\right)^{\frac{1}{2}}}\right)}-\frac{1}{2}\arctan\left(\frac{x}{2}\right) + C, C \in \mathbb{R}\) The integral of \( \frac{5}{(x-1)(x^2+4)} \) with respect to \( x \) is given by: \[ \int \frac{5}{(x-1)(x^2+4)} dx = \ln\left(\frac{|x-1|}{\sqrt{x^2+4}}\right) - \frac{1}{2}\arctan\left(\frac{x}{2}\right) + C \] where \( C \) is the constant of integration.