Question

lf $I={\int }_{-a}^a(\alpha {\sin }^{5}x+\beta {\tan }^{3}x+\gamma \cos x)dx$, 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

Answer: (A)

$\gamma ,a$

$I={\int }_{-a}^a(\alpha si{n}^{5}x+\beta ta{n}^{3}x+\gamma cosx)dx-\left(1\right)$
$I={\int }_{-a}^a\left[\alpha \right(si{n}^5(-x)+\beta ta{n}^3(-x)+\gamma cos(-x)\left)\right]dx$
$={\int }_{-a}^a[-\alpha si{n}^{5}x-\beta ta{n}^{3}x+\gamma cosx]dx-\left(2\right)$
(1)+(2)
$2I={\int }_{-a}^a\gamma cosxdx$
$I{=}^1{/}_2{\int }_{-a}^a\gamma cosxdx$
So, depends on a,$\gamma$