Nguyên hàm $I=\int{{\frac{{x\ln \left( {1+{{x}^{2}}} \right)+{{x}^{3}}}}{{1+{{x}^{2}}}}dx}}$ bằng?
A. $\frac{1}{4}{{\ln }^{2}}(1+{{x}^{2}})+\frac{1}{2}{{x}^{2}}-\ln \left( {1+{{x}^{2}}} \right)+C.$ 
B. $\frac{1}{4}{{\ln }^{2}}(1+{{x}^{2}})+\frac{1}{2}{{x}^{2}}-\frac{1}{2}\ln (1+{{x}^{2}})+C.$
C. $-\frac{1}{4}{{\ln }^{2}}(1+{{x}^{2}})+{{x}^{2}}-\frac{1}{2}\ln \left( {1+{{x}^{2}}} \right)+C.$ 
D. $-\frac{1}{4}{{\ln }^{2}}(1+{{x}^{2}})+\frac{1}{2}{{x}^{2}}-\ln \left( {1+{{x}^{2}}} \right)+C.$

Tìm nguyên hàm
$\int{{\frac{{\cot x}}{{{{{\sin }}^{2}}x}}dx}}.$
A. $-\frac{{{{{\cot }}^{2}}x}}{2}+C.$
B. $\frac{{{{{\cot }}^{2}}x}}{2}+C.$    
C. $-\frac{{{{{\tan }}^{2}}x}}{2}+C.$ 
D. $\frac{{{{{\tan }}^{2}}x}}{2}+C.$

 Họ nguyên hàm của hàm số $f(x)=\sqrt{x}+\sqrt[3]{x}+\sqrt[4]{x}$ là
A. $\frac{2}{3}\sqrt[2]{{{{x}^{3}}}}+\frac{3}{4}\sqrt[3]{{{{x}^{4}}}}+\frac{5}{4}\sqrt[4]{{{{x}^{5}}}}+C.$
B. $\frac{2}{3}\sqrt[2]{{{{x}^{3}}}}+\frac{3}{4}\sqrt[3]{{{{x}^{4}}}}+\frac{4}{5}\sqrt[4]{{{{x}^{5}}}}+C.$
C. $\frac{2}{3}\sqrt[2]{{{{x}^{3}}}}+\frac{4}{3}\sqrt[3]{{{{x}^{4}}}}+\frac{5}{4}\sqrt[4]{{{{x}^{5}}}}+C.$
D. $\frac{3}{2}\sqrt[2]{{{{x}^{3}}}}+\frac{4}{3}\sqrt[3]{{{{x}^{4}}}}+\frac{5}{4}\sqrt[4]{{{{x}^{5}}}}+C.$

Tìm nguyên hàm
$\int{{\frac{1}{{{{{\left( {5x-3} \right)}}^{2}}}}dx}}$
A. $-\frac{1}{{5\left( {5x-3} \right)}}+C.$       
B. $\frac{1}{{5\left( {5x-3} \right)}}+C.$  
C. $-\frac{1}{{\left( {5x-3} \right)}}+C.$  
D. $-\frac{1}{{5\left( {5x+3} \right)}}+C.$

Tìm nguyên hàm
$\int{{\frac{{{{e}^{{\sqrt{x}}}}}}{{\sqrt{x}}}}}dx.$
A. $2{{e}^{{\sqrt{x}}}}+C.$    
B. ${{e}^{{\sqrt{x}}}}+C.$   
C. $\frac{1}{2}{{e}^{{\sqrt{x}}}}+C.$  
D. $\frac{1}{{{{e}^{{\sqrt{x}}}}}}+C.$

Nguyên hàm $I=\int{{\frac{{\tan \left( {x-\frac{\pi }{4}} \right)}}{{\cos 2x}}dx}}$ bằng?
A. $I=\frac{1}{{2(tanx+1)}}+C.$ 
B. $I=\frac{1}{{2(tanx+1)}}.$ 
C. $I=\frac{1}{{1+\tan x}}+C.$ 
D. $I=\frac{1}{{1+\tan x}}.$

Xác định biểu thức $F\left( x \right)=\int{x\left( 1+\sin 2x \right)dx=a{{x}^{2}}+bx\cos 2x+c\sin 2x+C}.$ 
A. $\frac{{{x}^{2}}}{2}-x\frac{\cos 2x}{2}+\frac{\sin 2x}{4}+C.$
B. $-\frac{{{x}^{2}}}{2}+x\frac{\cos 2x}{2}-\frac{\sin 2x}{4}+C.$
C. $\frac{{{x}^{2}}}{2}-x\frac{\cos 2x}{2}-\frac{\sin 2x}{2}+C.$
D. $\frac{{{x}^{2}}}{2}+x\frac{\cos 2x}{2}+\frac{\sin 2x}{4}+C.$

Tìm nguyên hàm
$\displaystyle \int{{\frac{{{{{\cos }}^{2}}x}}{{{{{\sin }}^{4}}x}}}}dx.$
A. $\displaystyle -\frac{1}{3}{{\cot }^{3}}x+C.$ 
B. $\displaystyle \frac{1}{3}{{\cot }^{3}}x+C.$ 
C. $\displaystyle \frac{{5{{{\cos }}^{3}}x}}{{3{{{\sin }}^{5}}x}}+C.$
D. $A,B,C\begin{array}{*{20}{c}} {} & {sai.} \end{array}$

Cho hàm số f(x) = x3 – 2x2 + x
Gọi F(x) là nguyên hàm của hàm số f(x) thỏa mãn: $F(1)=\frac{1}{{12}}.$
Tìm F(x)?
A. $F(x)=\frac{{{{x}^{4}}}}{4}-\frac{2}{3}{{x}^{3}}+\frac{{{{x}^{2}}}}{2}.$
B. $F(x)=\frac{{{{x}^{4}}}}{4}-\frac{2}{3}{{x}^{3}}+\frac{{{{x}^{2}}}}{2}+C.$
C. $F(x)=\frac{{{{x}^{4}}}}{4}-\frac{2}{3}{{x}^{3}}+\frac{{{{x}^{2}}}}{2}+1.$
D. $F(x)=\frac{{{{x}^{4}}}}{4}-\frac{2}{3}{{x}^{3}}+\frac{{{{x}^{2}}}}{2}-2.$

Nguyên hàm $\int{2x{{e}^{x}}dx}$ bằng
A. $x{{e}^{x}}-2{{e}^{x}}+C.$
B. $2x{{e}^{x}}-{{e}^{x}}+C.$
C. $2x{{e}^{x}}+2{{e}^{x}}+C.$
D. $2x{{e}^{x}}-2{{e}^{x}}+C.$