25.
\(I=\int x\left(1-e^x\right)dx\)
Đặt \(\left\{{}\begin{matrix}u=x\\dv=\left(1-e^x\right)dx\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}du=dx\\v=x-e^x\end{matrix}\right.\)
\(\Rightarrow I=x\left(x-e^x\right)-\int\left(x-e^x\right)dx\)
\(=x\left(x-e^x\right)-\dfrac{x^2}{2}+e^x+C\)
\(=\dfrac{x^2}{2}+\left(1-x\right)e^x+C\)
26.
\(\int\limits^{13}_4f\left(x\right)dx=\int\limits^{13}_1f\left(x\right)dx-\int\limits^4_1f\left(x\right)dx=16-2=14\)
27.
\(\int\limits^2_1\dfrac{x^2-2x+5}{x}dx=\int\limits^2_1\left(x-2+\dfrac{5}{x}\right)dx\)
\(=\left(\dfrac{x^2}{2}-2x+5ln\left|x\right|\right)|^2_1\)
\(=\dfrac{-1}{2}+5ln2=\dfrac{1}{-2}+ln32\)
\(\Rightarrow a=-2;b=32\)
\(\Rightarrow a+b=30\)
28.
\(\int\limits^{-3}_1g\left(x\right)dx=-2\Rightarrow\int\limits^1_{-3}g\left(x\right)dx=2\)
\(I=\int\limits^1_{-3}5xdx+3\int\limits^1_{-3}f\left(x\right)dx+2\int\limits^1_{-3}g\left(x\right)dx\)
\(=-20+3.7+2.2=5\)
4 đáp án đều sai
29.
\(\int\limits^{\pi}_0cos^2x.sinxdx=-\int\limits^{\pi}_0cos^2x.d\left(cosx\right)\)
\(=-\dfrac{cos^3x}{3}|^{\pi}_0=\dfrac{2}{3}\)
30.
\(\int\limits^1_0\left(3m-2x\right)dx=\left(3m.x-x^2\right)|^1_0=3m-1\)
\(\Rightarrow3m-1=5\Rightarrow m=2\)
31.
\(I=\int\limits^1_0\left(x+1\right)e^xdx\)
Đặt \(\left\{{}\begin{matrix}u=x+1\\dv=e^xdx\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}du=dx\\v=e^x\end{matrix}\right.\)
\(\Rightarrow I=\left(x+1\right)e^x|^1_0-\int\limits^1_0e^xdx=2e-1-e^x|^1_0\)
\(=e\)
\(\Rightarrow\left\{{}\begin{matrix}a=0\\b=1\end{matrix}\right.\)
32.
\(\overrightarrow{u}=\left(7;-24;4\right)\)
33.
Mặt cầu tâm \(I\left(1;3;-2\right)\) bán kính \(R=\sqrt{1^2+3^2+\left(-2\right)^2-\left(-2010\right)}=\sqrt{2024}\)
34.
\(\overrightarrow{n}=\left(1;-2;-1\right)\)
