evaluate the following indefinite integral. ∫(10x^9 - 9/x^9) dx ∫(10x^9 - 9/x^9) dx = □

Answer:

$x^{10}+\frac{9}{8x^{8}}+C$

Explanation:

Step1: Split the integral

$\int(10x^{9}-\frac{9}{x^{9}})dx=\int10x^{9}dx-\int\frac{9}{x^{9}}dx$

Step2: Integrate the first - term

Using the power - rule for integration $\int x^{n}dx=\frac{x^{n + 1}}{n+1}+C$ ($n\neq - 1$), for $\int10x^{9}dx$, we have $10\times\frac{x^{9 + 1}}{9+1}=x^{10}$

Step3: Rewrite and integrate the second - term

Rewrite $\frac{9}{x^{9}}$ as $9x^{-9}$. Then $\int9x^{-9}dx=9\times\frac{x^{-9 + 1}}{-9 + 1}=\frac{9x^{-8}}{-8}=-\frac{9}{8x^{8}}$

Step4: Combine the results

$\int10x^{9}dx-\int\frac{9}{x^{9}}dx=x^{10}+\frac{9}{8x^{8}}+C$