Question

If $I=\underset{-a}{\overset{a}{\int }}(\alpha {\sin }^{5}x+\beta {\tan }^{3}x+\gamma \cos x)dx$, where $\alpha ,\beta ,\gamma$ are constants, then the value of $I$ depends on

• A. $\gamma ,a$
• B. $\alpha ,\beta ,\gamma ,a$
• C. $\alpha ,\beta ,a$
• D. $\beta$

Answer 1

The correct option is A $\gamma ,a$

We have,
$I=\underset{-a}{\overset{a}{\int }}(\alpha {\sin }^{5}x+\beta {\tan }^{3}x+\gamma \cos x)dx=\alpha \underset{-a}{\overset{a}{\int }}{\sin }^{5}xdx+\beta \underset{-a}{\overset{a}{\int }}{\tan }^{3}xdx+\gamma \underset{-a}{\overset{a}{\int }}\cos xdx=\alpha \times 0+\beta \times 0+\gamma \times 2\underset{0}{\overset{a}{\int }}\cos xdx=2\gamma \underset{0}{\overset{a}{\int }}\cos xdx\because {\sin }^{5}x\text{and}{\tan }^{3}x\text{are odd functions}$
$\therefore I$ depends on $\gamma ,a$