Đặt \( t = \sqrt{x} \Rightarrow x = t^2 \Rightarrow dx = 2t \, dt \).

\( \int x \sin \sqrt{x} \, dx = \int 2t^3 \sin t \, dt \)

Đặt: \( \begin{cases} u = t^3 \\ dv = \sin t \, dt \end{cases} \) (Tích phân từng phần 3 lần)

page90

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Đặt: \( \begin{cases} u = x \\ dv = \tan^2{x} \, dx \end{cases} \quad \Rightarrow \quad \begin{cases} du = dx \\ v = \tan{x} - x \end{cases} \)

\( I = x(\tan{x} - x) - \int (\tan{x} - x) \, dx \)

\(  = x(\tan{x} - x) - \int \left( \frac{\sin{x}}{\cos{x}} - x \right) \, dx \)

\(  = x(\tan{x} - x) + \ln|\cos{x}| + \frac{x^2}{2} + C. \)

page91

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Đặt: \( \begin{cases} u = x \\ dv = \frac{\sin{x}}{\cos^3{x}} \, dx \end{cases} \quad \Rightarrow \quad \begin{cases} du = dx \\ v = -\frac{1}{2\cos^2{x}} \end{cases} \)

\( \int \frac{x \sin{x}}{\cos^3{x}} \, dx = -\frac{x}{2 \cos^2{x}} + \frac{1}{2} \int \frac{1}{\cos^2{x}} \, dx \)

 \(  = -\frac{x}{2 \cos^2{x}} + \frac{1}{2} \tan{x} + C. \)

page92

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\(  \int \frac{x}{\cos^2{x}} \, dx + \int \frac{\sin{x}}{\cos^2{x}} \, dx \) 
\( \int \frac{\sin{x}}{\cos^2{x}} \, dx = \frac{1}{\cos{x}} + C \)
\( \int \frac{x}{\cos^2{x}} \, dx \)

Đặt: \( \begin{cases} u = x \\ dv = \frac{1}{\cos^2{x}} \, dx \end{cases} \quad \Rightarrow \quad \begin{cases} du = dx \\ v = \tan{x} \end{cases} \)

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

\(  = x \tan{x} + \ln|\cos{x}| + C \)

Vậy: 
\( \int \frac{x + \sin{x}}{\cos^2{x}} \, dx = x \tan{x} + \ln|\cos{x}| + \frac{1}{\cos{x}} + C. \)

page93

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Đặt: \( \begin{cases} u = x - 2 \\ dv = e^{2x} \, dx \end{cases} \quad \Rightarrow \quad \begin{cases} du = dx \\ v = \frac{1}{2} e^{2x} \end{cases} \)

\( I = \frac{(x - 2)}{2} e^{2x} - \int \frac{1}{2} e^{2x} \, dx \)
\( I = \frac{(x - 2)}{2} e^{2x} - \frac{1}{4} e^{2x} + C \)

page94 

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