Đang học Lý mà thấy bài nguyên hàm hay hay nên nhảy vô luôn :b

\(I_1=\int\limits^1_0xf\left(x\right)dx\)

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

\(\Rightarrow\int xf\left(x\right)dx=\dfrac{1}{2}x^2f\left(x\right)-\dfrac{1}{2}\int x^2f'\left(x\right)dx\)

\(\Rightarrow\int\limits^1_0xf\left(x\right)dx=\dfrac{1}{2}x^2|^1_0-\dfrac{1}{2}\int\limits^1_0x^2f'\left(x\right)dx=\dfrac{1}{5}\)

\(\Leftrightarrow\dfrac{1}{2}\int\limits^1_0\left[f'\left(x\right)\right]^2dx=\dfrac{3}{10}\Rightarrow\int\limits^1_0x^2f'\left(x\right)dx=\dfrac{3}{5}\)

Đoạn này hơi rối xíu, ông để ý kỹ nhé, nhận thấy ta có 2 dữ kiện đã biết, là: \(\int\limits^1_0\left[f'\left(x\right)\right]^2dx=\dfrac{9}{5}and\int\limits^1_0x^2f'\left(x\right)dx=\dfrac{3}{5}\) có gì đó liên quan đến hằng đẳng thức, nên ta sẽ sử dụng luôn

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

\(\Leftrightarrow\int\limits^1_0\left[f'\left(x\right)\right]^2dx+2t\int\limits^1_0x^2f'\left(x\right)dx+t^2\int\limits^1_0x^4dx=0\)

\(\Leftrightarrow\dfrac{9}{5}+\dfrac{6}{5}t+\dfrac{1}{5}t^2=0\)  \(\left(\int\limits^1_0x^4dx=\dfrac{1}{5}x^5|^1_0=\dfrac{1}{5}\right)\)\(\)\(\Leftrightarrow t=-3\Rightarrow\int\limits^1_0\left[f'\left(x\right)-3x^2\right]^2dx=0\)

\(\Leftrightarrow f'\left(x\right)=3x^2\Leftrightarrow f\left(x\right)=x^3+C\)

\(\Rightarrow\int\limits^1_0f\left(x\right)dx=\int\limits^1_0x^3dx=\dfrac{1}{4}x^4|^1_0=\dfrac{1}{4}\)

P/s: Có gì ko hiểu hỏi mình nhé !

Cho hàm số y=f(x)y=f(x) có đạo hàm và liên tục trên [0;π2][0;π2]thoả mãn f(x)=f′(x)−2cosxf(x)=f′(x)−2cosx. Biết f(π2)=1f(π2)=1, tính giá trị f(π3)f(π3)

A. √3+1/2         B. √3−1/2          C. 1−√3/2             D. 0

B

\(f'\left(x\right)-f\left(x\right)=2cosx\)

\(\Leftrightarrow e^{-x}.f'\left(x\right)-e^{-x}.f\left(x\right)=2e^{-x}cosx\)

\(\Rightarrow\left[e^{-x}.f\left(x\right)\right]'=2e^{-x}.cosx\)

Lấy nguyên hàm 2 vế:

\(\Rightarrow e^{-x}.f\left(x\right)=\int2e^{-x}cosxdx=e^{-x}\left(sinx-cosx\right)+C\)

Thay \(x=\dfrac{\pi}{2}\Rightarrow e^{-\dfrac{\pi}{2}}.1=e^{-\dfrac{\pi}{2}}+C\Rightarrow C=0\)

\(\Rightarrow f\left(x\right)=sinx-cosx\)

\(\Rightarrow f\left(\dfrac{\pi}{3}\right)=\dfrac{\sqrt{3}-1}{2}\)

a: f(-1/2)=17/4

f(5)=29

\(a,f\left(-\dfrac{1}{2}\right)=\dfrac{1}{4}+4=\dfrac{17}{4}\\ f\left(5\right)=25+4=29\\ b,f\left(x\right)=10=x^2+4\Leftrightarrow x^2=6\Leftrightarrow\left[{}\begin{matrix}x=\sqrt{6}\\x=-\sqrt{6}\end{matrix}\right.\)

\(1.x^2+\dfrac{1}{x^2}-2m\left(x+\dfrac{1}{x}\right)+1+2m=0\left(1\right)\)\(đặt:x^2+\dfrac{1}{x^2}=t\)

\(x>0\Rightarrow t\ge2\sqrt{x^2.\dfrac{1}{x^2}}=2\)

\(x< 0\Rightarrow-t=-x^2+\dfrac{1}{\left(-x^2\right)}\ge2\Rightarrow t\le-2\)

\(\Rightarrow t\in(-\infty;-2]\cup[2;+\infty)\left(2\right)\)

\(\Rightarrow\left(1\right)\Leftrightarrow t^2-2mt+2m-1=0\)

\(\Leftrightarrow\left(t-1\right)\left(t-2m+1\right)=0\Leftrightarrow\left[{}\begin{matrix}t=1\notin\left(2\right)\\t=2m-1\end{matrix}\right.\)

\(\Rightarrow\left[{}\begin{matrix}2m-1\le-2\\2m-1\ge2\end{matrix}\right.\)\(\Leftrightarrow\left[{}\begin{matrix}m\le-\dfrac{1}{2}\\m\ge\dfrac{3}{4}\end{matrix}\right.\)

\(2.\)  \(f^2\left(\left|x\right|\right)+\left(m-2\right)f\left(\left|x\right|\right)+m-3=0\left(1\right)\)

\(\Leftrightarrow\left[{}\begin{matrix}f\left(\left|x\right|\right)=-1\\f\left(\left|x\right|\right)=3-m\end{matrix}\right.\)

\(dựa\) \(vào\) \(đồ\) \(thị\) \(f\left(\left|x\right|\right)\) \(\Rightarrow f\left(\left|x\right|\right)=-1\) \(có\) \(2nghiem\) \(pb\)

\(\left(1\right)có\) \(6\) \(ngo\) \(pb\Leftrightarrow\left\{{}\begin{matrix}-1< 3-m< 3\\3-m\ne-1\\\end{matrix}\right.\)\(\Leftrightarrow0< m< 4\)

\(\Rightarrow m=\left\{1;2;3\right\}\)

Cho hàm số 

Ta có f(\(\dfrac{-3}{2}\)) = -3. (\(\dfrac{-3}{2}\))

                    = \(\dfrac{-3.\left(-3\right)}{2}\)

                    =\(\dfrac{9}{2}\)

Ta có f(-1) = -3. (-1)

                 = 3

Vậy f(\(\dfrac{-3}{2}\)) = \(\dfrac{9}{2}\) và f(-1) = 3.

a.

\(\Leftrightarrow x^2+2\left(m-1\right)x+m^2+3m+5\ne0\) ; \(\forall x\)

\(\Leftrightarrow\Delta'=\left(m-1\right)^2-\left(m^2+3m+5\right)< 0\)

\(\Leftrightarrow-5m-4< 0\)

\(\Leftrightarrow m>-\dfrac{4}{5}\)

b. 

\(\Leftrightarrow x^2+2\left(m-1\right)x+m^2+m-6\ge0\) ;\(\forall x\)

\(\Leftrightarrow\Delta'=\left(m-1\right)^2-\left(m^2+m-6\right)\le0\)

\(\Leftrightarrow-3m+7\le0\)

\(\Rightarrow m\ge\dfrac{7}{3}\)

c.

\(x^2-2\left(m+3\right)x+m+9>0\) ;\(\forall x\)

\(\Leftrightarrow\Delta'=\left(m+3\right)^2-\left(m+9\right)< 0\)

\(\Leftrightarrow m^2+5m< 0\Rightarrow-5< m< 0\)

a) \(f\left( 1 \right) = 3.1 = 3;f\left( { - 2} \right) = 3.\left( { - 2} \right) =  - 6;f\left( {\dfrac{1}{3}} \right) = 3.\dfrac{1}{3} = 1\).

b) Ta có: \(f\left( { - 3} \right) = 3.\left( { - 3} \right) =  - 9;f\left( { - 1} \right) = 3.\left( { - 1} \right) =  - 3\)

\(f\left( 0 \right) = 3.0 = 0;f\left( 2 \right) = 3.2 = 6;f\left( 3 \right) = 3.3 = 9\);

Ta lập được bảng sau