*Đặt tên các biểu thức theo thứ tự lần lượt là A,B,C,D,E,F *
Câu 1)
Ta có: \(d(\cos x)=(\cos x)'d(x)=-\sin xdx\)
\(\Rightarrow -d(\cos x)=\sin xdx\)
\(\Rightarrow A=\int \sqrt{3\cos x+2}\sin xdx=-\int \sqrt{3\cos x+2}d(\cos x)\)
Đặt \(\sqrt{3\cos x+2}=t\Rightarrow \cos x=\frac{t^2-2}{3}\)
\(\Rightarrow A=-\int td\left(\frac{t^2-2}{3}\right)=-\int t.\frac{2}{3}tdt=-\frac{2}{3}\int t^2dt=-\frac{2}{3}.\frac{t^3}{3}+c\)
\(=-\frac{2}{9}t^3+c=\frac{-2}{9}\sqrt{(3\cos x+2)^3}+c\)
Câu 2:
\(B=\int (1+\sin^3x)\cos xdx=\int \cos xdx+\int \sin ^3xcos xdx\)
\(=\int \cos xdx+\int \sin ^3xd(\sin x)\)
\(=\sin x+\frac{\sin ^4x}{4}+c\)
Câu 3:
\(C=\int \frac{e^x}{\sqrt{e^x-5}}dx=\int \frac{d(e^x)}{\sqrt{e^x-5}}\)
Đặt \(\sqrt{e^x-5}=t\Rightarrow e^x=t^2+5\)
Khi đó: \(C=\int \frac{d(t^2+5)}{t}=\int \frac{2tdt}{t}=\int 2dt=2t+c=2\sqrt{e^x-5}+c\)
Câu 4:
\(D=\int (x\sin x+2)dx=\int x\sin xdx+\int 2dx\)
Đặt \(\left\{\begin{matrix} u= x\\ dv= \sin xdx\end{matrix}\right.\Rightarrow \left\{\begin{matrix} du=dx\\ v=\int \sin xdx=-\cos x \end{matrix}\right.\)
\(\Rightarrow \int x\sin xdx= -x\cos x+\int \cos xdx=-x\cos x+\sin x+c\)
\(\int 2dx=2x+c\)
Do đó: \(D=-x\cos x+\sin x+2x+c\)
Câu 5:
\(E=\int 2x\cos xdx=2\int x\cos xdx\)
Đặt \(\left\{\begin{matrix} u=x\\ dv=\cos xdx\end{matrix}\right.\Rightarrow \left\{\begin{matrix} du=dx\\ v=\int \cos xdx=\sin x\end{matrix}\right.\)
\(\Rightarrow \int x\cos xdx=x\sin x-\int \sin xdx=x\sin x+\cos x+c\)
\(\Rightarrow E=2x\sin x+2\cos x+c\)
Câu 6:
\(\int 3^2\ln xdx=9\int \ln xdx\)
Đặt \(\left\{\begin{matrix} u=\ln x\\ dv=dx\end{matrix}\right.\Rightarrow \left\{\begin{matrix} du=\frac{dx}{x}\\ v=x\end{matrix}\right.\)
\(\Rightarrow \int \ln xdx=x\ln x-\int dx=x\ln x-x+c\)
\(\Rightarrow F=9x\ln x-9x+c\)
