Question
Evaluate: $\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} x^3 \sin ^4 x d x$
✓
Answer
$\text { Let } f ( x )= x ^3 \cdot \sin ^4 x$
$\therefore f (- x )=(- x )^3[\sin (- x )]^4$
$\therefore- x ^3(-\sin x )^4$
$\therefore- f ( x )$
$\therefore f ( x ) \text { is an odd function. }$
$\therefore \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} x^3 \cdot \sin ^4 x d x=0 \quad \ldots . . .\left[\because \int_{- a }^{ a } f (x) d x=0, \text { if } f ( x ) \text { is a odd function. }\right]$
$\therefore f (- x )=(- x )^3[\sin (- x )]^4$
$\therefore- x ^3(-\sin x )^4$
$\therefore- f ( x )$
$\therefore f ( x ) \text { is an odd function. }$
$\therefore \int_{-\frac{\pi}{4}}^{\frac{\pi}{4}} x^3 \cdot \sin ^4 x d x=0 \quad \ldots . . .\left[\because \int_{- a }^{ a } f (x) d x=0, \text { if } f ( x ) \text { is a odd function. }\right]$
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