a) \(I_1=\int\dfrac{dx}{x^2+2x+3}\)

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

\(=\dfrac{1}{\sqrt{2}}arctan\left(\dfrac{x+1}{\sqrt{2}}\right)+C\)

b) \(I_2=\int\dfrac{dx}{4x^2+4x+2}\)

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

\(=\dfrac{1}{2}arctan\left(2x+1\right)+C\)

a) \(I_4=\int\dfrac{3x+5}{2x^2+x+10}dx\)

\(=\int\dfrac{\dfrac{3}{4}\left(4x+1\right)+\dfrac{17}{4}}{2x^2+x+10}dx=\dfrac{3}{4}\int\dfrac{\left(4x+1\right)dx}{2x^2+x+10}+\dfrac{17}{4}\int\dfrac{dx}{2x^2+x+10}\)

\(=\dfrac{3}{4}\int\dfrac{d\left(2x^2+x+10\right)}{2x^2+x+10}+\dfrac{17}{8}\int\dfrac{dx}{x^2+\dfrac{x}{2}+5}\)

\(=\dfrac{3}{4}\ln\left(2x^2+x+10\right)+\dfrac{17}{8}\int\dfrac{dx}{\left(x+\dfrac{1}{4}\right)^2+\dfrac{79}{16}}\)

\(=\dfrac{3}{4}\ln\left(2x^2+x+10\right)+\dfrac{17}{8}\int\dfrac{dx}{\left(x+\dfrac{1}{4}\right)^2+\dfrac{79}{16}}\)

\(=\dfrac{3}{4}\ln\left(2x^2+x+10\right)+\dfrac{17}{8}\int\dfrac{d\left(x+\dfrac{1}{4}\right)}{\left(x+\dfrac{1}{4}\right)^2+\left(\dfrac{\sqrt{79}}{4}\right)^2}\)

\(=\dfrac{3}{4}\ln\left(2x^2+x+10\right)+\dfrac{17}{8}.\dfrac{4}{\sqrt{79}}arctan\left(\dfrac{4x+1}{\sqrt{79}}\right)+C\)

\(=\dfrac{3}{4}\ln\left(2x^2+x+10\right)+\dfrac{17}{2\sqrt{79}}arctan\left(\dfrac{4x+1}{\sqrt{79}}\right)+C\)

b) \(I_5=\int\dfrac{4x-1}{6x^2+9x+4}dx\)

\(=\int\dfrac{\dfrac{1}{3}\left(12x+9\right)-4}{6x^2+9x+4}dx\)

\(=\dfrac{1}{3}\int\dfrac{\left(12x+9\right)dx}{6x^2+9x+4}-4\int\dfrac{dx}{6x^2+9x+4}\)

\(=\dfrac{1}{3}\int\dfrac{d\left(6x^2+9x+4\right)}{6x^2+9x+4}-4\int\dfrac{dx}{\left(3x+1\right)^2+3}\)

\(=\dfrac{1}{3}\ln\left(6x^2+9x+4\right)-\dfrac{4}{3}\int\dfrac{d\left(3x+1\right)}{\left(3x+1\right)^2+\left(\sqrt{3}\right)^2}\)

\(=\dfrac{1}{3}\ln\left(6x^2+9x+4\right)-\dfrac{4}{3}.\dfrac{1}{\sqrt{3}}arctan\left(\dfrac{3x+1}{\sqrt{3}}\right)+C\)

4.

\(sinx+2cos\left(x+\dfrac{\pi}{3}\right)+4sin\left(x+\dfrac{\pi}{6}\right)+cosx=4\)

\(\Leftrightarrow sinx+cosx-\sqrt{3}sinx+2\sqrt{3}sinx+2cosx+cosx=4\)

\(\Leftrightarrow\left(1+\sqrt{3}\right)sinx+4cosx=4\)

\(\Leftrightarrow\sqrt{20+2\sqrt{3}}\left(\dfrac{1+\sqrt{3}}{\sqrt{20+2\sqrt{3}}}sinx+\dfrac{4}{\sqrt{20+2\sqrt{3}}}cosx\right)=4\)

\(\Leftrightarrow cos\left(x-arccos\dfrac{4}{\sqrt{20+2\sqrt{3}}}\right)=\dfrac{4}{\sqrt{20+2\sqrt{3}}}\)

\(\Leftrightarrow x-arccos\dfrac{4}{\sqrt{20+2\sqrt{3}}}=\pm arccos\dfrac{4}{\sqrt{20+2\sqrt{3}}}+k2\pi\)

\(\Leftrightarrow\left[{}\begin{matrix}x=2arccos\dfrac{4}{\sqrt{20+2\sqrt{3}}}+k2\pi\\x=k2\pi\end{matrix}\right.\)

3.

\(4sinx+cosx+2cos\left(x+\dfrac{\pi}{3}\right)=2\)

\(\Leftrightarrow4sinx+cosx+cosx-\sqrt{3}sinx=2\)

\(\Leftrightarrow\left(4-\sqrt{3}\right)sinx+2cosx=2\)

\(\Leftrightarrow\sqrt{23-4\sqrt{3}}\left(\dfrac{4-\sqrt{3}}{\sqrt{23-4\sqrt{3}}}sinx+\dfrac{2}{\sqrt{23-4\sqrt{3}}}cosx\right)=2\)

\(\Leftrightarrow cos\left(x-arccos\dfrac{2}{\sqrt{23-4\sqrt{3}}}\right)=\dfrac{2}{\sqrt{23-4\sqrt{3}}}\)

\(\Leftrightarrow x-arccos\dfrac{2}{\sqrt{23-4\sqrt{3}}}=\pm arccos\dfrac{2}{\sqrt{23-4\sqrt{3}}}+k2\pi\)

\(\Leftrightarrow\left[{}\begin{matrix}x=2arccos\dfrac{2}{\sqrt{23-4\sqrt{3}}}+k2\pi\\x=k2\pi\end{matrix}\right.\)

ĐK: \(x\ge0\)

Dễ thấy \(1-\sqrt{2\left(x^2-x+1\right)}\le1-\sqrt{2}< 0\)

Khi đó bất phương trình tương đương:

\(x-\sqrt{x}\le1-\sqrt{2\left(x^2-x+1\right)}\)

\(\Leftrightarrow\sqrt{x}-\dfrac{1}{\sqrt{x}}-1+\sqrt{2\left(x+\dfrac{1}{x}-1\right)}\le0\)

\(\Leftrightarrow\sqrt{x}-\dfrac{1}{\sqrt{x}}-1+\sqrt{2\left(\sqrt{x}-\dfrac{1}{\sqrt{x}}\right)^2+2}\le0\)

\(\Leftrightarrow t-1+\sqrt{2t^2+2}\le0\)

Khối lượng của hợp kim đó :

\(8,5:\dfrac{5}{8}=13,6\left(kg\right)\)

a: Xét ΔABE và ΔACD có

AB=AC

\(\widehat{BAE}\) chung

AE=AD

Do đó: ΔABE=ΔACD

b: Ta có: ΔABE=ΔACD

nên BE=CD

c: Xét ΔDBC và ΔECB có 

DB=EC

DC=EB

BC chung

Do đó: ΔDBC=ΔECB

Suy ra: \(\widehat{KCB}=\widehat{KBC}\)

hay ΔKBC cân tại K

d: Xét ΔABK và ΔACK có 

AB=AC

BK=CK

AK chung

Do đó: ΔABK=ΔACK

Suy ra: \(\widehat{BAK}=\widehat{CAK}\)

hay AK là tia phân giác của góc BAC