What is value of \(\rm \int_{-\pi/2}^{\pi/2} x^{5}\sin^{4}x dx\) ?

Concept: 

The rules for integrating even and odd functions ,

If the function is even or odd and the interval is [-a, a], we can apply these rules:

• When f(x) is even ⇔ \(\rm\int_{-a}^{a}f(x)dx= 2 \int_{0}^{a}f(x)dx\)
• When f(x) is odd ⇔ \(\rm\int_{-a}^{a}f(x)dx= 0\) 

​Calculation: 

Let f (x) = x⁵ sin⁴ x 

f (-x) = (-x)⁵ sin⁴ (-x) = -x⁵ (-sin x )⁴ = -x⁵ sin⁴ x .

⇒ f(-x) = - f(x) 

So, f(x) is an odd function . 

We knoiw that, when f(x) is odd ⇔ \(\rm\int_{-a}^{a}f(x)dx= 0\) 

Hence , \(\rm \int_{-\pi/2}^{\pi/2} x^{5}\sin^{4}x dx\) = 0 .  

The correct option is 4 .