Question

The value of integral $\int |5-4x|dx$ will be

• A. $5x-2x^2+C$ for $x\in [2,6]$
• B. $2x^2-5x+C$ for $x\in [2,6]$
• C. $2x^2-5x+C$ for $x\in [-2,-6]$
• D. $5x-2x^2+C$ for $x\in [-2,-6]$

Answer 1

Answer: (B), (D)

The correct options are
B $2x^2-5x+C$ for $x\in [2,6]$
D $5x-2x^2+C$ for $x\in [-2,-6]$

Let $I=\int |5-4x|dx$

$5-4x>0$

$5>4x$

$\frac{5}{4}>x$,

So $x\in (-\infty ,\frac{5}{4})$

And $x\in (\frac{5}{4},\infty )$ if $5-4x<0$

$\Rightarrow$ for $[2,6]$ $|5-4x|$ have $-ve$ sign

$\therefore I=\int (4x-5)dx$

$=\frac{4x^2}{2}-5x+C=2x^2-5x+C$

For $[-2,-6]$, $|5-4x|$ will have $+ve$ sign

So, just opposite signed.

Thus, $I=5x-2x^2+C$ for $x\in [-2,-6]$