213)
Let it be I3
doing f(a+b-x) we get 2I3=(same lims as given) ∫cos2x(xsinx+ cosx)2
but ∫cos2x(xsinx+ cosx)2 = sinxxsinx + cos x+ c (well known tric P(x)/q(x) rule)
\hspace{-16}(1):\; \mathbf{\int x\sqrt{1+\sin x}dx}\\\\\\ (2):\; \mathbf{\int_{0}^{\pi}\frac{2+2(x+1)\sin x-(x^2+1)\cos^2 x}{\sin x-x\cos x+1}dx}\\\\\\ (3):\;\mathbf{\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{\cos^2 x}{(e^x+1)(x\sin x+\cos x)^2}dx}\\\\\\
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2 Answers
2111)
I=\int x\sqrt{\left ( sin\frac{x}{2} +cos\frac{x}{2}\right )^2} dx
\int x{\left ( sin\frac{x}{2} +cos\frac{x}{2}\right )} dx
\int x\left sin\frac{x}{2}dx +\int xcos\frac{x}{2}\right dx
\texttt{using integration by parts we get }
\int x\left sin\frac{x}{2}dx =-2xcos\left (\frac{x}{2}\right )+ 4sin\left ( \frac{x}{2} \right )
\int x\left cos\frac{x}{2}dx =2xsin\left (\frac{x}{2}\right )+ 4cos\left ( \frac{x}{2} \right )
I =\int x\left cos\frac{x}{2}dx+\int x\left sin\frac{x}{2}dx=2xsin\left (\frac{x}{2}\right )+ 4cos\left ( \frac{x}{2} \right )-2xcos\left ( \frac{x}{2} \right )+4sin\left ( \frac{x}{2} \right )
\boxed{\boxed{I =2(x+2)sin\left (\frac{x}{2}\right ) -2(x-2)cos\left ( \frac{x}{2} \right )}}
