Cho hai hàm f(x) và g(x) có đạo hàm trên [1,2] thỏa mãn f(1)=g(1)=0 và x/(x+1)^2 g(x) +2017x=(x+1)f'(x)

Cho hàm số $f\left(x\right)=\left\{\begin{array}{ll}x\left(1+x^2\right)& khix\ge 3\\ \frac{1}{x-4}& khix<3\end{array}\right.$ . Tích phân $\underset{{\text{e}}^2}{\overset{{\text{e}}^4}{\int }}\frac{f\left(\ln x\right) }{x}\text{d}x$  bằng:

Cho hàm số $f\left(x\right)=\left\{\begin{array}{ll}\frac{1}{x}& khix\ge 1\\ x+1& khix<1\end{array}\right.$ . Tích phân $\underset{-2}{\overset{1}{\int }}f\left(\sqrt[3]{1-x}\right)\text{d}x=\frac{m}{n}$  ( $\frac{m}{n}$ là phân số tối giản), khi đó $m-2n$  bằng:

Cho hàm số $f\left(x\right)$  liên tục trên R và $\underset{0}{\overset{1}{\int }}f\left(x\right)\text{d}x=4$ , $\underset{0}{\overset{3}{\int }}f\left(x\right)\text{d}x=6$  . Tính $I=\underset{-1}{\overset{1}{\int }}f\left(\left|2x+1\right|\right)\text{d}x$

Cho hàm số $f\left(x\right)=\left\{\begin{array}{l}{x}^3+x+2\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}khi }x<1\\ x+3\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}khi }x\ge 1\end{array}\right.$ . Tính tích phân $\underset{0}{\overset{\frac{\pi }{2}}{\int }}f\left(3{\sin }^{2}x-1\right)\sin 2x\text{d}x$ .

Cho hàm số $f\left(x\right)=\left\{\begin{array}{l}2x-1\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}khi\hspace{0.17em}}x\ge 1\\ x^2\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}khi\hspace{0.17em}}x<1\end{array}\right.$ . Tính tích phân $\underset{1}{\overset{13}{\int }}f\left(\sqrt{x+3}-2\right)\text{d}x$ .

Cho hàm số $f\left(x\right)=\left\{\begin{array}{l}2x-4\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}khi\hspace{0.17em}}x\ge 2\\ 4-2x\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}khi\hspace{0.17em}}x<2\end{array}\right.$ . Tính tích phân $\underset{-\frac{\pi }{4}}{\overset{\frac{\pi }{2}}{\int }}f\left(3-4{\cos }^{2}x\right)\sin 2x\text{d}x$ .

Cho hàm số $f\left(x\right)=\left\{\begin{array}{l}{x}^4+2x^2-1\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}khi }x<1\\ 3-x^2\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}khi }x\ge 1\end{array}\right.$ . Tính tích phân $\underset{1}{\overset{e^4}{\int }}f\left(\sqrt{4-\ln x}\right)\frac{1}{x}\text{d}x$ .