find ∫(5/x² + 7x + 2)dx ∫(5/x² + 7x + 2)dx = + c question help: video written example submit question…

Explanation:

Step1: Split the integral

Use the sum - rule of integration $\int(f(x)+g(x)+h(x))dx=\int f(x)dx+\int g(x)dx+\int h(x)dx$. So, $\int\left(\frac{5}{x^{2}}+7x + 2\right)dx=\int\frac{5}{x^{2}}dx+\int7xdx+\int2dx$.

Step2: Rewrite and integrate $\int\frac{5}{x^{2}}dx$

Rewrite $\frac{5}{x^{2}}$ as $5x^{-2}$. Then, by the power - rule of integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), $\int5x^{-2}dx=5\int x^{-2}dx=5\times\frac{x^{-2 + 1}}{-2+1}=- \frac{5}{x}$.

Step3: Integrate $\int7xdx$

Using the power - rule $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ with $n = 1$, $\int7xdx=7\int xdx=7\times\frac{x^{1+1}}{1 + 1}=\frac{7}{2}x^{2}$.

Step4: Integrate $\int2dx$

Since $\int kdx=kx + C$ (where $k$ is a constant), $\int2dx=2x$.

Step5: Combine the results

$\int\left(\frac{5}{x^{2}}+7x + 2\right)dx=-\frac{5}{x}+\frac{7}{2}x^{2}+2x + C$.

Answer:

$-\frac{5}{x}+\frac{7}{2}x^{2}+2x + C$