Tính tích phân :
\(\int\limits^3_1\frac{3+\ln x}{\left(x+1\right)^2}dx\)
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Những câu hỏi liên quan
5 tháng 4 2016
Ta có \(I=\int\limits^{\frac{\pi}{3}}_{\frac{\pi}{4}}\frac{\ln2.\ln\left(2\tan x\right)}{\sin2x.\ln\left(2\tan x\right)}dx=\ln2\int\limits^{\frac{\pi}{3}}_{\frac{\pi}{4}}\frac{dx}{\sin2x.\ln\left(2\tan x\right)}+\int\limits^{\frac{\pi}{3}}_{\frac{\pi}{4}}\frac{dx}{\sin2x}\)
Tính \(\ln2\int\limits^{\frac{\pi}{3}}_{\frac{\pi}{4}}\frac{dx}{\sin2x.\ln\left(2\tan x\right)}=\frac{\ln2}{2}\int\limits^{\frac{\pi}{3}}_{\frac{\pi}{4}}\frac{d\left[\ln\left(2\tan x\right)\right]}{\ln2\left(2\tan x\right)}=\frac{\ln2}{2}\left[\ln\left(\ln\left(2\tan x\right)\right)\right]|^{\frac{\pi}{3}}_{\frac{\pi}{4}}=\frac{\ln2}{2}.\ln\left(\frac{\ln2\sqrt{3}}{\ln2}\right)\)
Tính \(\int\limits^{\frac{\pi}{3}}_{\frac{\pi}{4}}\frac{dx}{\sin2x}=\frac{1}{2}\ln\left(\tan x\right)|^{\frac{\pi}{3}}_{\frac{\pi}{4}}=\frac{1}{2}\ln\sqrt{3}\)
Vậy \(I=\frac{\ln2}{2}\ln\left(\frac{\ln2\sqrt{3}}{\ln2}\right)+\frac{1}{2}\ln\sqrt{3}\)
JC
1
NV
Nguyễn Việt Lâm
Giáo viên
28 tháng 2 2023
\(I=\int\limits^e_1xlnxdx+\int\limits^e_1\dfrac{lnx}{x}dx=I_1+I_2\)
Xét \(I_1\) , đặt \(\left\{{}\begin{matrix}u=lnx\\dv=xdx\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}du=\dfrac{dx}{x}\\v=\dfrac{x^2}{2}\end{matrix}\right.\)
\(\Rightarrow I_1=\dfrac{x^2}{2}lnx|^e_1-\int\limits^e_1\dfrac{x}{2}=\dfrac{e^2}{2}-\dfrac{e}{2}+\dfrac{1}{2}\)
Xét \(I_2=\int\limits^e_1\dfrac{lnx}{x}dx=\int\limits^e_1lnx.d\left(lnx\right)=\dfrac{ln^2x}{2}|^e_1=\dfrac{1}{2}\)
\(\Rightarrow I=\dfrac{e^2}{2}-\dfrac{e}{2}+1\)
NV
Nguyễn Việt Lâm
Giáo viên
11 tháng 4 2019
Câu 1:
\(\int\limits^3_0\left(f'\left(x\right)+1\right)\sqrt{x+1}dx=\int\limits^3_0f'\left(x\right)\sqrt{x+1}dx+\int\limits^3_0\sqrt{x+1}dx\)
\(=\int\limits^3_0f'\left(x\right)\sqrt{x+1}dx+\frac{14}{3}=\frac{302}{15}\Rightarrow\int\limits^1_0f'\left(x\right)\sqrt{x+1}dx=\frac{232}{15}\)
Ta có:
\(I=\int\limits^3_0\frac{f\left(x\right)dx}{\sqrt{x+1}}\)
Đặt \(\left\{{}\begin{matrix}u=f\left(x\right)\\dv=\frac{dx}{\sqrt{x+1}}\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}du=f'\left(x\right)dx\\v=2\sqrt{x+1}\end{matrix}\right.\)
\(\Rightarrow I=2f\left(x\right)\sqrt{x+1}|^3_0-2\int\limits^3_0f'\left(x\right)\sqrt{x+1}dx\)
\(=4f\left(3\right)-2f\left(0\right)-2.\frac{232}{15}\)
\(=2\left(2f\left(3\right)-f\left(0\right)\right)-\frac{464}{15}=36-\frac{464}{15}=\frac{76}{15}\)
NV
Nguyễn Việt Lâm
Giáo viên
11 tháng 4 2019
Câu 2:
\(I_1=\int\limits^3_1\frac{xf'\left(x\right)}{x+1}dx=0\)
Đặt \(\left\{{}\begin{matrix}u=\frac{x}{x+1}\\dv=f'\left(x\right)dx\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}du=\frac{1}{\left(x+1\right)^2}dx\\v=f\left(x\right)\end{matrix}\right.\)
\(\Rightarrow I_1=\frac{xf\left(x\right)}{x+1}|^3_1-\int\limits^3_1\frac{f\left(x\right)}{\left(x+1\right)^2}=\frac{3.3}{3+1}-\frac{1.3}{1+1}-\int\limits^3_1\frac{f\left(x\right)}{\left(x+1\right)^2}dx=\frac{3}{4}-\int\limits^3_1\frac{f\left(x\right)}{\left(x+1\right)^2}dx=0\)
\(\Rightarrow\int\limits^3_1\frac{f\left(x\right)}{\left(x+1\right)^2}dx=\frac{3}{4}\)
Ta có:
\(I=\int\limits^3_1\frac{f\left(x\right)+lnx}{\left(x+1\right)^2}dx=\int\limits^3_1\frac{f\left(x\right)}{\left(x+1\right)^2}dx+\int\limits^3_1\frac{lnx}{\left(x+1\right)^2}dx=\frac{3}{4}+I_2\)
Xét \(I_2=\int\limits^3_1\frac{lnx}{\left(x+1\right)^2}dx\Rightarrow\) đặt \(\left\{{}\begin{matrix}u=lnx\\dv=\frac{1}{\left(x+1\right)^2}\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}du=\frac{dx}{x}\\v=\frac{-1}{x+1}\end{matrix}\right.\)
\(\Rightarrow I_2=\frac{-lnx}{x+1}|^3_1+\int\limits^3_1\frac{dx}{x\left(x+1\right)}=-\frac{1}{4}ln3+\int\limits^1_0\left(\frac{1}{x}-\frac{1}{x+1}\right)dx\)
\(=-\frac{1}{4}ln3+ln\left(\frac{x}{x+1}\right)|^3_1=-\frac{1}{4}ln3+ln\frac{3}{4}-ln\frac{1}{2}=\frac{3}{4}ln3-ln2\)
\(\Rightarrow I=\frac{3}{4}+\frac{3}{4}ln3-ln2\)
NT
1
6 tháng 4 2016
\(I=\frac{1}{4}\int\limits^e_1\frac{4\ln^2x-1+1}{x\left(1+2\ln x\right)}dx=\frac{1}{4}\int\limits^e_1\frac{\left(2\ln x-1\right)dx}{x}+\frac{1}{4}\int\limits^e_1\frac{dx}{x\cdot\left(1+2\ln x\right)}\)
\(=\frac{1}{8}\int\limits^e_1\left(2\ln x-1\right)d\left(2\ln x-1\right)+\frac{1}{8}\int\limits^e_1\frac{d\left(2\ln x+1\right)}{\left(1+2\ln x\right)}\)
\(=\left(\frac{1}{16}\left(2\ln x-1\right)^2\right)|^e_1+\frac{1}{8}\ln\left|\left(1+2\ln x\right)\right||^e_1\)
\(=\frac{1}{8}\ln3\)
VB
6 tháng 4 2016
\(I=\int\limits^5_1\left(\frac{x}{\sqrt{x-1}+1}+\frac{\ln x}{\left(x+1\right)^2}\right)dx=\int\limits^5_1\frac{x}{\sqrt{x-1}+1}dx+\int\limits^5_1\frac{\ln x}{\left(x+1\right)^2}dx\)
- Tính \(\int\limits^5_1\frac{x}{\sqrt{x-1}+1}dx\)
Đặt \(t=\sqrt{x-1}\Rightarrow t^2=x-1\Leftrightarrow x=t^2+1\Rightarrow dx=2tdt\)
Đổi cận : Cho x=1 => t=0; x=5=>t=2
\(I_1=\int\limits^2_0\frac{t^2+1}{t+1}.2td=\int\limits^2_0\frac{2t^3+2t}{t+1}dt=\int\limits^2_0\left(2t^2-2t+4-\frac{4}{t+1}\right)dt\)
\(=\left(\frac{2}{3}t^3-t^2+4t-4\ln\left|x+1\right|\right)|^2_0=\frac{28}{3}-4\ln3\)
\(I_2=\int\limits^5_1\frac{\ln x}{\left(x+1\right)^2}dx\)
Đặt \(\begin{cases}u=\ln x\\dv=\frac{1}{\left(x+1\right)^2}dx\end{cases}\) \(\Rightarrow\begin{cases}du=\frac{1}{x}dx\\v=-\frac{1}{x+1}\end{cases}\)
Ta có \(I_2=-\frac{1}{x+1}\ln x|^5_1+\int\limits^5_1\frac{1}{x\left(x+1\right)}dx=-\frac{1}{6}\ln5+\int\limits^5_1\left(\frac{1}{x}-\frac{1}{x+1}\right)dx\)
\(=-\frac{1}{6}\ln5+\left(\ln\left|x\right|x+1\right)|^5_1=-\frac{1}{6}\ln5+\ln5-\ln6+\ln2=\frac{5}{6}\ln5-\ln3\)
Khi đó \(I=I_1+I_2=\frac{28}{3}+\frac{5}{6}\ln5=5\ln3\)
AH
Akai Haruma
Giáo viên
6 tháng 3 2017
Câu 1)
Ta có \(I=\int ^{1}_{0}\frac{dx}{\sqrt{3+2x-x^2}}=\int ^{1}_{0}\frac{dx}{4-(x-1)^2}\).
Đặt \(x-1=2\cos t\Rightarrow \sqrt{4-(x-1)^2}=\sqrt{4-4\cos^2t}=2|\sin t|\)
Khi đó:
\(I=\int ^{\frac{2\pi}{3}}_{\frac{\pi}{2}}\frac{d(2\cos t+1)}{2\sin t}=\int ^{\frac{2\pi}{3}}_{\frac{\pi}{2}}\frac{2\sin tdt}{2\sin t}=\int ^{\frac{2\pi}{3}}_{\frac{\pi}{2}}dt=\left.\begin{matrix} \frac{2\pi}{3}\\ \frac{\pi}{2}\end{matrix}\right|t=\frac{\pi}{6}\)
Câu 3)
\(K=\int ^{3}_{2}\ln (x^3-3x+2)dx=\int ^{3}_{2}\ln [(x+2)(x-1)^2]dx\)
\(=\int ^{3}_{2}\ln (x+2)d(x+2)+2\int ^{3}_{2}\ln (x-1)d(x-1)\)
Xét \(\int \ln tdt\): Đặt \(\left\{\begin{matrix} u=\ln t\\ dv=dt\end{matrix}\right.\Rightarrow \left\{\begin{matrix} du=\frac{dt}{t}\\ v=t\end{matrix}\right.\Rightarrow \int \ln t dt=t\ln t-t\)
\(\Rightarrow K=\left.\begin{matrix} 3\\ 2\end{matrix}\right|(x+2)[\ln (x+2)-1]+2\left.\begin{matrix} 3\\ 2\end{matrix}\right|(x-1)[\ln (x-1)-1]\)
\(=5\ln 5-4\ln 4-1+4\ln 2-2=5\ln 5-4\ln 2-3\)
AH
Akai Haruma
Giáo viên
6 tháng 3 2017
Bài 2)
\(J=\int ^{1}_{0}x\ln (2x+1)dx\). Đặt \(\left\{\begin{matrix} u=\ln (2x+1)\\ dv=xdx\end{matrix}\right.\Rightarrow \left\{\begin{matrix} du=\frac{2dx}{2x+1}\\ v=\frac{x^2}{2}\end{matrix}\right.\)
Khi đó:
\(J=\left.\begin{matrix} 1\\ 0\end{matrix}\right|\frac{x^2\ln (2x+1)}{2}-\int ^{1}_{0}\frac{x^2}{2x+1}dx\)\(=\frac{\ln 3}{2}-\frac{1}{4}\int ^{1}_{0}(2x-1+\frac{1}{2x+1})dx\)
\(=\frac{\ln 3}{2}-\left.\begin{matrix} 1\\ 0\end{matrix}\right|\frac{x^2-x}{4}-\frac{1}{8}\int ^{1}_{0}\frac{d(2x+1)}{2x+1}=\frac{\ln 3}{2}-\left.\begin{matrix} 1\\ 0\end{matrix}\right|\frac{\ln (2x+1)}{8}\)
\(=\frac{\ln 3}{2}-\frac{\ln 3}{8}=\frac{3\ln 3}{8}\)