Question

Let $I={\int }_{-a}^a(p{\tan }^{3}x+q{\cos }^{2}x+r\sin x)$ dx, where $p,q,r$ are arbitrary constants.

The numerical value of $I$ depends on?

• A. $p,q,r,a$
• B. $q,r,a$
• C. $q,a$
• D. $p,r,a$

Answer 1

The correct option is C $q,a$

$I={\int }_{-a}^a(p+{\tan }^{3}x+q{\cos }^{2}x+r\sin x)dx$

$={\int }_{-a}^{a}p{\tan }^{3}xdx+{\int }_{-a}^{a}q{\cos }^{2}x+{\int }_{-a}^{a}r\sin xdx$
As $p{\tan }^{3}x$ and $r\sin x$ are odd function
$\therefore {\int }_{-a}^{a}p{\tan }^{3}xdx=0$ and ${\int }_{-a}^{a}r\sin xdx=0$
$\therefore I={\int }_{-a}^{a}q{\cos }^{2}xdx={\int }_{-a}^{a}q(\frac{1}{2}+\frac{{\cos }^{2}x}{2})dx$
$=q{[\frac{1}{2}x+\frac{\sin 2x}{4}]}_{-a}^a$
Therefore $I$ depends on $q,a$