Cho hàm số f(x)= 3x+3 khi x< 1/2 và x+4 khi x>= 1/2 . Tính tích phân bi/2 đến 0 của f(sinx) cosxdx.

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

Cho hàm số $f\left(x\right)=\left\{\begin{array}{l}1-x^2\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}khi }x\le 1\\ 2x-2\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}khi }x>1\end{array}\right.$ . Tính tích phân $\underset{-\frac{\pi }{2}}{\overset{\frac{\pi }{4}}{\int }}f\left(5\sin 2x-1\right)\cos 2x\text{d}x$ .

Cho hàm số $f\left(x\right)=\left\{\begin{array}{l}2x^3-x-5\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}khi\hspace{0.17em}}x\ge 2\\ 11-x\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}khi\hspace{0.17em}}x<2\end{array}\right.$ . Tính tích phân $\underset{\frac{1}{e}}{\overset{e}{\int }}f\left(2+\ln x\right)\frac{1}{x}\text{d}x$ .

Cho hàm số $f\left(x\right)=\left\{\begin{array}{l}1-x^2\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}khi\hspace{0.33em}}x\le 3\\ 7-5x\text{\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}khi\hspace{0.17em}\hspace{0.17em}}x>3\end{array}\right.$ . Tính tích phân $\underset{0}{\overset{\ln 2}{\int }}f\left(3e^x-1\right){\text{e}}^x\text{d}x$  .