Let $$I = \int_{ - a}^a {\left( {p{{\tan }^3}x + q{{\cos }^2}x + r\sin \,x} \right)dx,} $$ 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 :
$$q,\,a$$
Solution :
$$I = p\int_{ - a}^a {{{\tan }^3}x\,dx} + q\int_{ - a}^a {{{\cos }^2}x\,dx} + r\int_{ - a}^a {\sin \,x\,dx} $$
$$ = p \times 0 + 2q\int_0^a {{{\cos }^2}x\,dx} + r \times 0,$$ because $${\tan ^3}x$$ and $$\sin \,x$$ are odd functions, and $${\cos ^2}x$$ is an even function.
Releted MCQ Question on Calculus >> Application of Integration
Releted Question 1
The area bounded by the curves $$y = f\left( x \right),$$ the $$x$$-axis and the ordinates $$x = 1$$ and $$x = b$$ is $$\left( {b - 1} \right)\sin \left( {3b + 4} \right).$$ Then $$f\left( x \right)$$ is-