Solveeit Logo
Login

Question

Question: \[\int_{- \pi/2}^{\pi/2}{\sin^{4}x\cos^{6}xdx =}\]...

∫−π/2π/2sin⁡4xcos⁡6xdx=\int_{- \pi/2}^{\pi/2}{\sin^{4}x\cos^{6}xdx =}

A

3π64\frac{3\pi}{64}

B

3π572\frac{3\pi}{572}

C

3π256\frac{3\pi}{256}

D

3π128\frac{3\pi}{128}

Answer

3π256\frac{3\pi}{256}

Explanation

Solution

I=∫−π/2π/2sin⁡4xcos⁡6xdx=2∫0π/2sin⁡4xcos⁡6x.dxI = \int_{- \pi/2}^{\pi/2}{\sin^{4}x\cos^{6}xdx}\mathbf{= 2}\int_{\mathbf{0}}^{\mathbf{\pi/2}}{\mathbf{\sin}^{\mathbf{4}}\mathbf{x}\mathbf{\cos}^{\mathbf{6}}\mathbf{x}\mathbf{.dx}}

∵∫−aaf(x)dx=2∫0af(x)dx,if f(−x)=f(x)=0,if f(−x)=−f(x)\begin{matrix} \because\int_{- a}^{a}{f(x)dx = 2\int_{0}^{a}{f(x)dx,}} & \text{if }f( - x) = f(x) \\ = 0, & \text{if }f( - x) = - f(x) \end{matrix}

Applying Gamma function, we get

I=2Γ5/2.Γ7/22.Γ6\mathbf{I =}\frac{\mathbf{2}\mathbf{\Gamma}\mathbf{5/2.}\mathbf{\Gamma}\mathbf{7/2}}{\mathbf{2.}\mathbf{\Gamma}\mathbf{6}}

=3/2.1/2.π.5/2.3/2.1/2.π5.4.3.2.1=3π28=3π256\mathbf{=}\frac{\mathbf{3/2.1/2.}\sqrt{\mathbf{\pi.}}\mathbf{5/2.3/2.1/2.}\sqrt{\mathbf{\pi}}}{\mathbf{5.4.3.2.1}}\mathbf{=}\frac{\mathbf{3\pi}}{\mathbf{2}^{\mathbf{8}}}\mathbf{=}\frac{\mathbf{3\pi}}{\mathbf{256}}.