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Question: The value of \(\int_{0}^{\pi/2}{\sin^{4}x\cos^{6}xdx}\) =...

The value of ∫0π/2sin⁡4xcos⁡6xdx\int_{0}^{\pi/2}{\sin^{4}x\cos^{6}xdx} =

A

3π⥂/⥂3123\pi ⥂ / ⥂ 312

B

5π/5125\pi/512

C

3π/5123\pi/512

D

5π/3125\pi/312

Answer

3π/5123\pi/512

Explanation

Solution

I=(4−1).(4−3).(6−1).(6−3).(6−5)(4+6)(4+6−2)(4+6−4)(4+6−6)(4+6−8).π2=3.1.5.3.110.8.6.4.2..π2=3π512I = \frac{(4 - 1).(4 - 3).(6 - 1).(6 - 3).(6 - 5)}{(4 + 6)(4 + 6 - 2)(4 + 6 - 4)(4 + 6 - 6)(4 + 6 - 8)}.\frac{\pi}{2} = \frac{3.1.5.3.1}{10.8.6.4.2.}.\frac{\pi}{2} = \frac{3\pi}{512}

(1) ∫0∞e−axsin⁡bxdx=ba2+b2\int_{0}^{\infty}{e^{- ax}\sin bxdx} = \frac{b}{a^{2} + b^{2}}

(2) ∫0∞e−axcos⁡bxdx=aa2+b2\int_{\mathbf{0}}^{\mathbf{\infty}}{\mathbf{e}^{\mathbf{-}\mathbf{ax}}\mathbf{\cos}\mathbf{b}\mathbf{xdx}}\mathbf{=}\frac{\mathbf{a}}{\mathbf{a}^{\mathbf{2}}\mathbf{+}\mathbf{b}^{\mathbf{2}}}

(3) ∫0∞e−axxndx=n!an+1\int_{0}^{\infty}e^{- ax}x^{n}dx = \frac{n!}{a^{n} + 1}