evaluate the indefinite integral.\n int \frac{5x + 4}{x^{2}+1}dx \n(it helps to realize that (\frac{5x +…

Explanation:

Step1: Split the integral

We split $\int\frac{5x + 4}{x^{2}+1}dx$ into $\int\frac{5x}{x^{2}+1}dx+\int\frac{4}{x^{2}+1}dx$.

Step2: Solve $\int\frac{5x}{x^{2}+1}dx$

Let $u = x^{2}+1$, then $du=2xdx$. So $\int\frac{5x}{x^{2}+1}dx=\frac{5}{2}\int\frac{du}{u}=\frac{5}{2}\ln|u|=\frac{5}{2}\ln|x^{2}+1|$.

Step3: Solve $\int\frac{4}{x^{2}+1}dx$

We know that $\int\frac{1}{x^{2}+1}dx=\arctan(x)+C$, so $\int\frac{4}{x^{2}+1}dx = 4\arctan(x)$.

Step4: Combine the results

$\int\frac{5x + 4}{x^{2}+1}dx=\frac{5}{2}\ln|x^{2}+1|+4\arctan(x)+C$.

Answer:

$\frac{5}{2}\ln|x^{2}+1|+4\arctan(x)+C$