Question

1. write the functions in the brackets

(a) xdx = d( )
(c) n ≠ - 1, x^n dx = d( )
(e) 1/x dx = d( )

Explanation:

Step1: Recall power - rule for integration

The antiderivative of $x^{n}$ is $\frac{x^{n + 1}}{n+1}+C$ when $n
eq - 1$. For $x=x^{1}$, using the power - rule with $n = 1$, we have $\int xdx=\frac{1}{2}x^{2}+C$, so $x dx=d(\frac{1}{2}x^{2}+C)$.

Step2: General power - rule case

For a general non - negative integer $n
eq - 1$, $\int x^{n}dx=\frac{1}{n + 1}x^{n+1}+C$, so $x^{n}dx=d(\frac{1}{n + 1}x^{n+1}+C)$.

Step3: Antiderivative of $\frac{1}{x}$

The antiderivative of $\frac{1}{x}$ is $\ln|x|+C$, so $\frac{1}{x}dx=d(\ln|x|+C)$.

Answer:

A. $x dx = d(\frac{1}{2}x^{2}+C)$; C. $n
eq - 1,x^{n}dx=d(\frac{1}{n + 1}x^{n+1}+C)$; E. $\frac{1}{x}dx=d(\ln|x|+C)$