Question

Consider the following statements :
I) ${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\sqrt{\cos x-{\cos }^{3}x}dx=\frac{3}{4}.$

II) ${\int }_0^4\left(\left|x-1\right|+\left|x-3\right|\right)dx=10.$
Which of these is are correct

• A. only(I)
• B. only (II)
• C. Both (I) & (II)
• D. None of these

Answer 1

Answer: (B)

only (II)

$I={\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\sqrt{\cos x-{\cos }^{3}x}dx=\frac{3}{4}$

$I={\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\cos x(1-{\cos }^{2}x)dx$

${\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\sin x\sqrt{\cos x}dx[{\int }_a^{b}f\left(x\right)dx={\int }_a^{b}f(a+b-x)dx]$

$I={\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\sin (-x)\sqrt{\cos (-x)}dx=-{\int }_{-\frac{\pi }{2}}^{\frac{\pi }{2}}\sin x\sqrt{\cos x}dx$

$I=-I\Rightarrow 2I=0$

is $I=0$

$\therefore$ $I$ is false

(ii) ${\int }_0^4(|x-1|+|x-3|)dx=10$

$x\in [0,1]$

$|x-1|+|x-3|=4-2x$

$x\in [1,3]$

$|x+1|+|x-3|=2$

$x\in [3,4]$

$|x-1|+|x-3|=2x-4$

${\int }_0^4|x-1|+|x-3|dx={\int }_0^1(4-2x)dx+{\int }_1^{3}2dx+{\int }_3^4(2x-4)dx$

$={[4x-x^2]}_0^1+{\left[2x\right]}_1^3+{[x^2-4x]}_3^4=4-1+2(3-1)+(4^2-3^2)-4=3+4+7-4=10$

$\therefore$ is $II$ is true