1)
Đặt \(t=sin^2x\) cho gọn
\(dt=d\left(sin^2x\right)=2sinx\cdot cosx\cdot dx\)
Khi đó:
\(I_1=\int e^t\cdot\left(1-t^2\right)\cdot\frac{dt}{2}=\frac{1}{2}\int\left(1-t^2\right)d\left(e^t\right)\\ =\frac{1}{2}\cdot e^t\cdot\left(1-t^2\right)-\frac{1}{2}\int e^t\cdot d\left(1-t^2\right)=\frac{1}{2}\cdot e^t\cdot\left(1-t^2\right)+\int t\cdot e^t\cdot dt\\ =\frac{e^t\left(1-t^2\right)}{2}+t\cdot e^t-e^t+C\\ =e^t\left(-\frac{t^2}{2}+t-1\right)=...\)
ai giúp em giải bài 43 con 1 với con 4 với ạ. e cảm ơn ạ
4)
Đặt \(\sqrt{lnx+1}=t\Rightarrow lnx=t^2-1\)
\(d\left(lnx\right)=d\left(t^2-1\right)\Leftrightarrow\frac{dx}{x}=2t\cdot dt\)
Khi đó:
\(I_4=\int\frac{\left(t^2-1\right)^2}{t}\cdot2t\cdot dt\\ =2\int\left(t^4-2t^2+1\right)dt\\ =\frac{2}{5}t^5-\frac{4}{3}t^3+2t+C\\ =\frac{2}{5}\left(\sqrt{lnx+1}\right)^5-\frac{4}{3}\left(\sqrt{lnx+1}\right)^3+2\sqrt{lnx+1}+C\)
