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![](https://rs.olm.vn/images/avt/0.png?1311)
2.
\(I=\int e^{3x}.3^xdx\)
Đặt \(\left\{{}\begin{matrix}u=3^x\\dv=e^{3x}dx\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}du=3^xln3dx\\v=\dfrac{1}{3}e^{3x}\end{matrix}\right.\)
\(\Rightarrow I=\dfrac{1}{3}e^{3x}.3^x-\dfrac{ln3}{3}\int e^{3x}.3^xdx=\dfrac{1}{3}e^{3x}.3^x-\dfrac{ln3}{3}.I\)
\(\Rightarrow\left(1+\dfrac{ln3}{3}\right)I=\dfrac{1}{3}e^{3x}.3^x\)
\(\Rightarrow I=\dfrac{1}{3+ln3}.e^{3x}.3^x+C\)
1.
\(I=\int\left(2x-1\right)e^{\dfrac{1}{x}}dx=\int2x.e^{\dfrac{1}{x}}dx-\int e^{\dfrac{1}{x}}dx\)
Xét \(J=\int2x.e^{\dfrac{1}{x}}dx\)
Đặt \(\left\{{}\begin{matrix}u=e^{\dfrac{1}{x}}\\dv=2xdx\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}du=-\dfrac{e^{\dfrac{1}{x}}}{x^2}dx\\v=x^2\end{matrix}\right.\)
\(\Rightarrow J=x^2.e^{\dfrac{1}{x}}+\int e^{\dfrac{1}{x}}dx\)
\(\Rightarrow I=x^2.e^{\dfrac{1}{x}}+C\)
![](https://rs.olm.vn/images/avt/0.png?1311)
\(f\left(1-x\right)+f\left(x\right)=\dfrac{9^{1-x}}{9^{1-x}+3}+\dfrac{9^x}{9^x+3}=\dfrac{9}{9+3.9^x}+\dfrac{9^x}{9^x+3}=\dfrac{3}{9^x+3}+\dfrac{9^x}{9^x+3}=1\)
\(\Rightarrow f\left(x\right)=1-f\left(1-x\right)\)
\(\Rightarrow f\left(cos^2x\right)=1-f\left(sin^2x\right)\)
Do đó:
\(f\left(3m+\dfrac{1}{4}sinx\right)+f\left(cos^2x\right)=1\)
\(\Leftrightarrow f\left(3m+\dfrac{1}{4}sinx\right)=f\left(sin^2x\right)\) (1)
Hàm \(f\left(x\right)=\dfrac{9^x}{9^x+3}\) có \(f'\left(x\right)=\dfrac{3.9^x.ln9}{\left(9^x+3\right)^2}>0\Rightarrow f\left(x\right)\) đồng biến trên R
\(\Rightarrow\left(1\right)\Leftrightarrow3m+\dfrac{1}{4}sinx=sin^2x\)
Đến đây chắc dễ rồi, biện luận để pt \(sin^2x-\dfrac{1}{4}sinx=3m\) có 8 nghiệm trên khoảng đã cho
a) Điều kiện x>0. Thực hiện chia tử cho mẫu ta được:
f(x) =
=
= ![This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.](http://latex.codecogs.com/gif.latex?x%5E%7B%5Cfrac%7B2%7D%7B3%7D%7D+%20x%5E%7B%5Cfrac%7B1%7D%7B6%7D%7D%20+%20x%5E%7B-%5Cfrac%7B1%7D%7B3%7D%7D)
∫f(x)dx = ∫(
)dx =
+C
b) Ta có f(x) =
=
-e-x
; do đó nguyên hàm của f(x) là:
F(x)=
=
=
+ C
c) Ta có f(x) =![This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.](http://latex.codecogs.com/gif.latex?%5Cfrac%7B1%7D%7Bsin%5E%7B2%7Dx.cos%5E%7B2%7Dx%7D%3D%5Cfrac%7B4%7D%7Bsin%5E%7B2%7D2x%7D)
hoặc f(x) =![This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.](http://latex.codecogs.com/gif.latex?%5Cfrac%7B1%7D%7Bcos%5E%7B2%7Dx.sin%5E%7B2%7Dx%7D%3D%5Cfrac%7B1%7D%7Bcos%5E%7B2%7Dx%7D+%5Cfrac%7B1%7D%7Bsin%5E%7B2%7Dx%7D)
Do đó nguyên hàm của f(x) là F(x)= -2cot2x + C
d) Áp dụng công thức biến tích thành tổng:
f(x) =sin5xcos3x =
(sin8x +sin2x).
Vậy nguyên hàm của hàm số f(x) là F(x) = -
(
cos8x + cos2x) +C
e) ta có![This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.](http://latex.codecogs.com/gif.latex?tan%5E%7B2%7Dx%20%3D%20%5Cfrac%7B1%7D%7Bcos%5E%7B2%7Dx%7D-1)
vậy nguyên hàm của hàm số f(x) là F(x) = tanx - x + C
g) Ta có ∫e3-2xdx= -
∫e3-2xd(3-2x)= -
e3-2x +C
h) Ta có :![This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.](http://latex.codecogs.com/gif.latex?%5Cint%20%5Cfrac%7Bdx%7D%7B%281+x%29%281-2x%29%29%7D%3D%5Cfrac%7B1%7D%7B3%7D%5Cint%20%28%5Cfrac%7B1%7D%7B1+x%7D+%5Cfrac%7B2%7D%7B1-2x%7D%29dx)
=
= ![This is the rendered form of the equation. You can not edit this directly. Right click will give you the option to save the image, and in most browsers you can drag the image onto your desktop or another program.](http://latex.codecogs.com/gif.latex?%5Cfrac%7B1%7D%7B3%7Dln%5Cleft%20%7C%20%5Cfrac%7B1+x%7D%7B1-2x%7D%20%5Cright%20%7C%20+C)