Question
Download Solution PDFWhat is value of \(\rm \int_{-\pi/2}^{\pi/2} x^{5}\sin^{4}x dx\) ?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFConcept:
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 .
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