\( I = \int x \left( 3x + \sqrt{9x^2 - 1} \right) \, dx = \int \left( 3x^2 + x \sqrt{9x^2 - 1} \right) \, dx \)

\( = x^3 + \frac{1}{18} \cdot \frac{\left( 9x^2 - 1 \right)}{\frac{3}{2}}^{\frac{3}{2}} + C = x^3 + \frac{1}{27} \left( 9x^2 - 1 \right)^{\frac{3}{2}} + C  \Rightarrow \boxed{\text{C}} \)

page70

\( I = \int \frac{\cos x - \sin x}{(\sin x + \cos x)^2} \, dx \quad \) Đặt \(u = \sin x + \cos x) \)

\( = -\frac{1}{\sin x + \cos x} + C \)

page71

\( \int \frac{\cos 2x}{(\sin x + \cos x + 2)^2} \, dx = \int \frac{(\cos x - \sin x)(\cos x + \sin x)}{(\sin x + \cos x + 2)^2} \, dx \)

Đặt \( t = \sin x + \cos x + 2 \), suy ra \( dt = (\cos x - \sin x) \, dx \).

\( I = \int \frac{t - 2}{t^2} \, dt = \int \left( \frac{1}{t} - \frac{2}{t^2} \right) \, dt \)

\( = \ln |t| + \frac{2}{t} + C \)

page72

\( \int \left( \frac{\sin x - \cos x}{\sin x + \cos x} \right)^2 \, dx = \int \tan^2 \left( x - \frac{\pi}{4} \right) \, dx \quad\) Đặt\( \, t = x - \frac{\pi}{4} \)

page73 

\( \ln(e x) = 1 + \ln x \)

\( (3 + x \ln x)' = \ln x + 1 \)

\( \int \frac{\ln(e x)}{3 + x \ln x} \, dx = \ln |3 + x \ln x| + C \)

\( (1 + x e^x)' = (1 + x)e^x \)

\( \int \frac{(1+x)e^x}{1 + x e^x} \, dx = \ln |1 + x e^x| + C \)

page74