\(5;;\sqrt{\left(x+5\right)\left(3x+4\right)}>4\left(x-1\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}\left\{{}\begin{matrix}4\left(x-1\right)\le0\\\left(x+5\right)\left(3x+4\right)\ge0\end{matrix}\right.\\\left\{{}\begin{matrix}4\left(x-1\right)\ge0\\\left(x+5\right)\left(3x+4\right)\ge0\\\left(x+5\right)\left(3x+4\right)>16\left(x-1\right)^2\end{matrix}\right.\end{matrix}\right.\)

\(TH:\left\{{}\begin{matrix}4\left(x-1\right)\le0\\\left(x+5\right)\left(3x+4\right)\ge0\end{matrix}\right.\Leftrightarrow\left\{{}\begin{matrix}x\le1\\\left[{}\begin{matrix}x\le-5\\x\ge-\dfrac{4}{3}\end{matrix}\right.\end{matrix}\right.\)

\(\Leftrightarrow x\in(-\infty;-5]\cup\left[-\dfrac{4}{3};1\right]\left(1\right)\)

\(TH:\left\{{}\begin{matrix}4\left(x-1\right)\ge0\\\left(x+5\right)\left(3x+4\right)\ge0\\\left(x+5\right)\left(3x+4\right)>16\left(x-1\right)^2\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\ge1\\\left[{}\begin{matrix}x\le-5\\x\ge-\dfrac{4}{3}\end{matrix}\right.\\-\dfrac{1}{13}< x< 4\\\end{matrix}\right.\)\(\Rightarrow x\in[1;4)\left(2\right)\)

\(\left(1\right)\left(2\right)\Rightarrow x\in(-\infty;5]\cup[\dfrac{-4}{3};4)\)

\(6;;;;\sqrt{7x+7}+\sqrt{7x-6}+2\sqrt{49x^2+7x-42}< 181-14x\)

(đoạn 49x^2+7x+42 chắc bạn viết sai đề dấu"-" thành "+")

\(đk:\left\{{}\begin{matrix}7x+7\ge0\\7x-6\ge0\end{matrix}\right.\) \(\Leftrightarrow x\ge\dfrac{6}{7}\)

\(bpt\Leftrightarrow\sqrt{7x+7}+\sqrt{7x-6}+2\sqrt{\left(7x+7\right)\left(7x-6\right)}+14x+1< 182\left(1\right)\)

\(đặt:\sqrt{7x+7}+\sqrt{7x-6}=t>0\)

\(\Rightarrow t^2=14x+1+2\sqrt{\left(7x+7\right)\left(7x-6\right)}\)

\(\Rightarrow\left(1\right)\Leftrightarrow t^2+t< 182\Leftrightarrow-14< t< 13\)

\(\Rightarrow\sqrt{7x+7}+\sqrt{7x-6}< 13\Leftrightarrow14x+1+2\sqrt{\left(7x+7\right)\left(7x-6\right)}< 169\)

\(\Leftrightarrow2\sqrt{\left(7x+7\right)\left(7x-6\right)}< 168-14x\)

\(\Leftrightarrow\left\{{}\begin{matrix}168-14x\ge0\\\left(7x+7\right)\left(7x-6\right)\ge0\\4\left(7x+7\right)\left(7x-6\right)< \left(168-14x\right)^2\\\end{matrix}\right.\)

\(\Leftrightarrow\left\{{}\begin{matrix}x\le12\\\left[{}\begin{matrix}x\le-1\\x\ge\dfrac{6}{7}\end{matrix}\right.\\x< 6\\\end{matrix}\right.\)\(\Rightarrow\dfrac{6}{7}\le x< 6\)

\(7;\) \(3\sqrt{x}+\dfrac{3}{2\sqrt{x}}< 2x+\dfrac{1}{2x}-1\left(đk:x>0\right)\)

\(\Leftrightarrow3\left(\sqrt{x}+\dfrac{1}{2\sqrt{x}}\right)< 2\left(x+\dfrac{1}{4x}\right)-1\left(1\right)\)

\(đặt:\sqrt{x}+\dfrac{1}{2\sqrt{x}}=t>0\)

\(\Leftrightarrow t^2=\sqrt{x}^2+2.\sqrt{x}.\dfrac{1}{2\sqrt{x}}+\left(\dfrac{1}{2\sqrt{x}}\right)^2=x+\dfrac{1}{4x}+1\)

\(\Rightarrow x+\dfrac{1}{4x}=t^2-1\)

\(\left(1\right)\Leftrightarrow3t< 2\left(t^2-1\right)-1\)

\(\Leftrightarrow2t^2-3t-3>0\Leftrightarrow\left[{}\begin{matrix}t< \dfrac{3-\sqrt{33}}{4}\\t>\dfrac{3+\sqrt{33}}{4}\end{matrix}\right.\)

\(\Rightarrow\sqrt{x}+\dfrac{1}{2\sqrt{x}}>\dfrac{3+\sqrt{33}}{4}\)

\(\Leftrightarrow\dfrac{2x+1}{2\sqrt{x}}>\dfrac{3+\sqrt{33}}{4}\)

\(\Leftrightarrow\sqrt{x}< \dfrac{2\left(2x+1\right)}{3+\sqrt{33}}\Leftrightarrow\left\{{}\begin{matrix}x>0\\2\left(2x+1\right)\ge0\\x< \left[\dfrac{2\left(2x+1\right)}{3+\sqrt{33}}\right]^2\\\end{matrix}\right.\)

đến đây dễ dàng rồi như mấy ý trên bạn tự giải quyết để tìm ra x

\(I=\int\dfrac{2}{2+5sinxcosx}dx=\int\dfrac{2sec^2x}{2sec^2x+5tanx}dx\\ =\int\dfrac{2sec^2x}{2tan^2x+5tanx+2}dx\)

We substitute :

\(u=tanx,du=sec^2xdx\\ I=\int\dfrac{2}{2u^2+5u+2}du\\ =\int\dfrac{2}{2\left(u+\dfrac{5}{4}\right)^2-\dfrac{9}{8}}du\\ =\int\dfrac{1}{\left(u+\dfrac{5}{4}\right)^2-\dfrac{9}{16}}du\\ \)

Then, 

\(t=u+\dfrac{5}{4}\\I=\int\dfrac{1}{t^2-\dfrac{9}{16}}dt\\ =\int\dfrac{\dfrac{2}{3}}{t-\dfrac{3}{4}}-\dfrac{\dfrac{2}{3}}{t+\dfrac{3}{4}}dt\)

Finally,

\(I=\dfrac{2}{3}ln\left(\left|\dfrac{t-\dfrac{3}{4}}{t+\dfrac{3}{4}}\right|\right)+C=\dfrac{2}{3}ln\left(\left|\dfrac{tanx+\dfrac{1}{2}}{tanx+2}\right|\right)+C\)

2'31s

Gọi khối lượng giấy lớp 7A1 là a

khối lượng giấy lớp 7A2 là b

Áp dụng tính chất của dãy tỉ số bằng nhau, ta được:

\(\dfrac{a}{3}=\dfrac{b}{4}=\dfrac{b-a}{4-3}=23\)

Do đó: a=69; b=92

Theo định lí Pytago tam giác ABC vuông tại A

\(AC=\sqrt{BC^2-AB^2}=8cm\)

Vì AD là pg nên \(\frac{BD}{DC}=\frac{AB}{AC}\Rightarrow\frac{BD}{AB}=\frac{DC}{AC}\)

Theo tc dãy tỉ số bằng nhau 

\(\frac{BD}{AB}=\frac{DC}{AC}=\frac{BC}{AB+AC}=\frac{10}{14}=\frac{5}{7}\Rightarrow BD=\frac{30}{7}cm;CD=\frac{40}{7}cm\)