Question

find the indefinite integral and check your result by differentiation. (remember the constant of integration.)
int(csc (x) cot (x)-8 x) d x

Explanation:

Step1: Integrate term - by - term

Use the integral formulas \(\int\csc(x)\cot(x)dx=-\csc(x)+C_1\) and \(\int x^n dx=\frac{x^{n + 1}}{n+1}+C_2(n
eq - 1)\).
\(\int(\csc(x)\cot(x)-8x)dx=\int\csc(x)\cot(x)dx-\int8x dx\)
\(=-\csc(x)-8\times\frac{x^{2}}{2}+C\)

Step2: Simplify the expression

Simplify \(8\times\frac{x^{2}}{2}\) to \(4x^{2}\).
\(=-\csc(x)-4x^{2}+C\)

Step3: Check by differentiation

Differentiate \(y =-\csc(x)-4x^{2}+C\) using the differentiation formulas \((\csc(x))^\prime=-\csc(x)\cot(x)\) and \((x^n)^\prime=nx^{n - 1}\).
\(y^\prime=-(-\csc(x)\cot(x))-4\times2x+0\)
\(=\csc(x)\cot(x)-8x\)

Answer:

\(-\csc(x)-4x^{2}+C\)