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

  1. 1
  2. π/2 
  3. π/4 
  4. 0

Answer (Detailed Solution Below)

Option 4 : 0
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Detailed Solution

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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 )= -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 .

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