Lời giải:
a)
\(I=\int \frac{dx}{x+\sqrt{x}}\). Đặt \(\sqrt{x}=t\Rightarrow x=t^2\)
\(\Rightarrow I=\int \frac{d(t^2)}{t^2+t}=\int \frac{2tdt}{t^2+t}=2\int\frac{dt}{t+1}=2\int \frac{d(t+1)}{t+1}\)
\(\Leftrightarrow I=2\ln |t+1|+c=2\ln |\sqrt{x}+1|+c\)
b)
\(P=\int \sin 4x\cos 3x\sin xdx\)
Áp dụng công thức lượng giác:
\(\sin 4x-\sin 2x=2\cos 3x.\sin x\)
\(\Rightarrow P=\frac{1}{2}\int \sin 4x(\sin 4x-\sin 2x)dx\)
\(=\frac{1}{2}\int \sin ^24xdx-\frac{1}{2}\int \sin 4x\sin 2xdx\)
\(=\frac{1}{2}\int \frac{1-\cos 8x}{2}dx-\frac{1}{2}\int \frac{\cos 6x-\cos 2x}{-2}dx\)
\(=\frac{1}{4}\int (1-\cos 8x)dx+\frac{1}{4}\int (\cos 6x-\cos 2x)dx\)
\(=\frac{1}{4}x-\frac{\sin 8x}{32}+\frac{\sin 6x}{24}-\frac{\sin 2x}{8}+c\)
