Question

1. find the indefinite integral of the following:

$f(x) = 180x^{8} + 18x^{5} - 14x + 16 - \frac{5}{x} + 9\sec x\tan x$

Explanation:

Step1: Apply sum/difference integration rule

$\int f(x)dx = \int 180x^8 dx + \int 18x^5 dx - \int 14x dx + \int 16 dx - \int \frac{5}{x} dx + \int 9\sec x \tan x dx$

Step2: Integrate power terms (power rule)

For $\int ax^n dx = \frac{a}{n+1}x^{n+1} + C$:

• $\int 180x^8 dx = 180 \cdot \frac{x^{9}}{9} = 20x^9$
• $\int 18x^5 dx = 18 \cdot \frac{x^{6}}{6} = 3x^6$
• $\int 14x dx = 14 \cdot \frac{x^{2}}{2} = 7x^2$
• $\int 16 dx = 16x$

Step3: Integrate reciprocal term

$\int \frac{5}{x} dx = 5\ln|x|$

Step4: Integrate trigonometric term

$\int \sec x \tan x dx = \sec x$, so $\int 9\sec x \tan x dx = 9\sec x$

Step5: Combine all terms + constant

Add all results and include the constant of integration $C$.

Answer:

$20x^9 + 3x^6 - 7x^2 + 16x - 5\ln|x| + 9\sec x + C$