1.

\(I=\int\dfrac{cot^2x}{sin^6x}dx=\int\dfrac{cot^2x}{sin^4x}.\dfrac{1}{sin^2x}=\int cot^2x\left(1+cot^2x\right)^2.\dfrac{1}{sin^2x}dx\)

Đặt \(u=cotx\Rightarrow du=-\dfrac{1}{sin^2x}dx\)

\(I=-\int u^2\left(1+u^2\right)^2du=-\int\left(u^6+2u^4+u^2\right)du\)

\(=-\dfrac{1}{7}u^7+\dfrac{2}{5}u^5+\dfrac{1}{3}u^3+C\)

\(=-\dfrac{1}{7}cot^7x+\dfrac{2}{5}cot^5x+\dfrac{1}{3}cot^3x+C\)

2.

\(I=\int\left(e^{sinx}+cosx\right).cosxdx=\int e^{sinx}.cosxdx+\int cos^2xdx\)

\(=\int e^{sinx}.d\left(sinx\right)+\dfrac{1}{2}\int\left(1+cos2x\right)dx\)

\(=e^{sinx}+\dfrac{1}{2}x+\dfrac{1}{4}sin2x+C\)

\(\int e^x\left(2-x\right)dx\)

\(\left\{{}\begin{matrix}u=2-x\\dv=e^xdx\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}du=-dx\\v=e^x\end{matrix}\right.\)

\(\Rightarrow\int e^x\left(2-x\right)dx=e^x\left(2-x\right)+\int e^xdx=e^x\left(2-x\right)+e^x\)

\(\left\{{}\begin{matrix}u=ln\left(x+1\right)\\dv=xdx\end{matrix}\right.\Rightarrow\left\{{}\begin{matrix}du=\dfrac{dx}{x+1}\\v=\dfrac{1}{2}x^2\end{matrix}\right.\)

\(\Rightarrow\int xln\left(x+1\right)dx=\dfrac{1}{2}x^2.ln\left(x+1\right)-\dfrac{1}{2}\int\dfrac{x^2}{x+1}dx\)

\(\int\dfrac{x^2dx}{x+1}=\int\left(x-1\right)dx+\int\dfrac{dx}{x+1}\)

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Đặt \(2x+2=u\Rightarrow2xdx=du\Rightarrow dx=\dfrac{1}{2}du\)

\(\left\{{}\begin{matrix}x=0\Rightarrow u=2\\x=2\Rightarrow u=6\end{matrix}\right.\)

\(\Rightarrow I=\int\limits^6_2f\left(u\right).\dfrac{1}{2}du=\dfrac{1}{2}\int\limits^6_2f\left(u\right)du=\dfrac{1}{2}\int\limits^6_2f\left(x\right)dx=\dfrac{1}{2}.6=3\)

\(\int\dfrac{lnx}{x\left(2ln^2x-1\right)^3}dx\)

\(t=2ln^2x-1\Rightarrow dt=\dfrac{4}{x}lnxdx\Rightarrow dx=\dfrac{x.dt}{4lnx}\)

\(\Rightarrow\int\dfrac{lnx}{x\left(2ln^2x-1\right)^3}dx=\int\dfrac{lnx}{x\left(2ln^2x-1\right)^3}.\dfrac{xdt}{4lnx}=\dfrac{1}{4}\int\dfrac{dt}{t^3}=\dfrac{1}{4}.\left(-\dfrac{1}{2}\right).t^{-2}=-\dfrac{1}{8\sqrt{2ln^2x-1}}\)

\(\int\limits^3_{-1}f\left(\left|x\right|\right)dx=\int\limits^0_{-1}f\left(\left|x\right|\right)dx+\int\limits^1_0f\left(\left|x\right|\right)dx+\int\limits^3_1f\left(\left|x\right|\right)dx\)

\(=\int\limits^0_{-1}f\left(-x\right)dx+\int\limits^1_0f\left(x\right)dx+\int\limits^3_1f\left(x\right)dx\)

\(=\int\limits^1_0f\left(x\right)dx+\int\limits^1_0f\left(x\right)dx+\int\limits^3_1f\left(x\right)dx\)

\(=3+3+6=12\)

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Xét \(I=\int\limits^1_0x^2f\left(x\right)dx\)

Đặt \(\left\{{}\begin{matrix}u=f\left(x\right)\\dv=x^2dx\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}du=f'\left(x\right)dx\\v=\dfrac{1}{3}x^3\end{matrix}\right.\)

\(\Rightarrow I=\dfrac{1}{3}x^3.f\left(x\right)|^1_0-\dfrac{1}{3}\int\limits^1_0x^3.f'\left(x\right)dx=-\dfrac{1}{3}\int\limits^1_0x^3f'\left(x\right)dx\)

\(\Rightarrow\int\limits^1_0x^3f'\left(x\right)dx=-1\)

Lại có: \(\int\limits^1_0x^6.dx=\dfrac{1}{7}\)

\(\Rightarrow\int\limits^1_0\left[f'\left(x\right)\right]^2dx+14\int\limits^1_0x^3.f'\left(x\right)dx+49.\int\limits^1_0x^6dx=0\)

\(\Rightarrow\int\limits^1_0\left[f'\left(x\right)+7x^3\right]^2dx=0\)

\(\Rightarrow f'\left(x\right)+7x^3=0\)

\(\Rightarrow f'\left(x\right)=-7x^3\)

\(\Rightarrow f\left(x\right)=\int-7x^3dx=-\dfrac{7}{4}x^4+C\)

\(f\left(1\right)=0\Rightarrow C=\dfrac{7}{4}\)

\(\Rightarrow I=\int\limits^1_0\left(-\dfrac{7}{4}x^4+\dfrac{7}{4}\right)dx=...\)