Hướng dẫn giải Bài 3 (Trang 113 SGK Toán Giải tích 12)

<p><strong>Bài 3 (Trang 113 SGK Toán Giải tích 12):</strong></p> <p>Sử dụng phương pháp biến đổi số, tính tích phân sau</p> <p>a) <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mn>3</mn></msubsup><mfrac><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><msup><mrow><mo>(</mo><mn>1</mn><mo>-</mo><mi mathvariant="normal">x</mi><mo>)</mo></mrow><mstyle displaystyle="true"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle></msup></mfrac><mi>dx</mi></math> (đặt u = x + 1);</p> <p>b) <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><msqrt><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></msqrt><mi>dx</mi></math>  (đặt x = sin t);</p> <p>c) <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mfrac><mrow><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup><mo>(</mo><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi><mo>)</mo></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>xe</mi><mi mathvariant="normal">x</mi></msup></mrow></mfrac><mi>dx</mi></math>  (đặtu = 1 + x.e<sup>x</sup>);</p> <p>d) <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mfrac><mi mathvariant="normal">a</mi><mn>2</mn></mfrac></msubsup><mfrac><mn>1</mn><msqrt><msup><mi mathvariant="normal">a</mi><mn>2</mn></msup><mo>-</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></msqrt></mfrac><mi>dx</mi></math> (a>0) (đặt x=asin t).</p> <p><strong><em>Hướng dẫn giải:</em></strong></p> <p>a) Đặt u = x + 1 => du = dx</p> <table style="border-collapse: collapse; width: 20.3655%; height: 21px;" border="1"> <tbody> <tr> <td style="width: 19.8472%;">x</td> <td style="width: 80.2517%;">0          3</td> </tr> <tr> <td style="width: 19.8472%;">u</td> <td style="width: 80.2517%;">1          4</td> </tr> </tbody> </table> <p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mn>3</mn></msubsup><mfrac><mrow><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mi>dx</mi></mrow><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi><mo>)</mo><mstyle displaystyle="true"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle></mrow></mfrac><mo> </mo><mo>=</mo><mo> </mo><msubsup><mo>∫</mo><mn>1</mn><mn>4</mn></msubsup><mfrac><msup><mrow><mo>(</mo><mi mathvariant="normal">u</mi><mo>-</mo><mn>1</mn><mo>)</mo></mrow><mn>2</mn></msup><msup><mi mathvariant="normal">u</mi><mstyle displaystyle="true"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle></msup></mfrac><mi>du</mi><mo> </mo><mo> </mo><mo>=</mo><mo> </mo><msubsup><mo>∫</mo><mn>1</mn><mn>4</mn></msubsup><mfrac><mrow><msup><mi mathvariant="normal">u</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi mathvariant="normal">u</mi><mo>+</mo><mn>1</mn></mrow><msup><mi mathvariant="normal">u</mi><mstyle displaystyle="true"><mfrac><mn>3</mn><mn>2</mn></mfrac></mstyle></msup></mfrac><mi>du</mi><mo> </mo><mo>=</mo><mo> </mo><msubsup><mo>∫</mo><mn>1</mn><mn>4</mn></msubsup><mo>(</mo><msup><mi mathvariant="normal">u</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><msup><mi mathvariant="normal">u</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo>+</mo><msup><mi mathvariant="normal">u</mi><mrow><mo>-</mo><mfrac><mn>3</mn><mn>2</mn></mfrac></mrow></msup><mo>)</mo><mi>du</mi><mo> </mo><mspace linebreak="newline"/><mo>=</mo><mo> </mo><mo>(</mo><mfrac><mn>2</mn><mn>3</mn></mfrac><msup><mi mathvariant="normal">u</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mo>-</mo><mn>4</mn><msup><mi mathvariant="normal">u</mi><mfrac><mn>1</mn><mn>2</mn></mfrac></msup><mo>-</mo><mn>2</mn><msup><mi mathvariant="normal">u</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></msup><mo>)</mo><msubsup><mo>|</mo><mn>1</mn><mn>4</mn></msubsup><mo> </mo><mspace linebreak="newline"/><mo>=</mo><mo> </mo><mfrac><mn>5</mn><mn>3</mn></mfrac></math></p> <p>b) Đặt x = sin t => dx = costdt</p> <p>Đổi cận:</p> <table style="border-collapse: collapse; width: 17.624%; height: 57px;" border="1"> <tbody> <tr> <td style="width: 21.9458%;">x</td> <td style="width: 77.5013%;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mn>1</mn></math></td> </tr> <tr> <td style="width: 21.9458%;">t</td> <td style="width: 77.5013%;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfrac><mi>π</mi><mn>2</mn></mfrac></math></td> </tr> </tbody> </table> <p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><msqrt><mn>1</mn><mo>-</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></msqrt><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><msubsup><mo>∫</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup><msqrt><mn>1</mn><mo>-</mo><msup><mi>sin</mi><mn>2</mn></msup><mi mathvariant="normal">t</mi></msqrt><mo>.</mo><mi>costdt</mi><mo> </mo><mo>=</mo><mo> </mo><msubsup><mo>∫</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup><msup><mi>cos</mi><mn>2</mn></msup><mi>tdt</mi><mo> </mo><mo>=</mo><mo> </mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msubsup><mo>∫</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup><mo>(</mo><mn>1</mn><mo>+</mo><mi>cos</mi><mn>2</mn><mi mathvariant="normal">t</mi><mo>)</mo><mi>dt</mi><mo> </mo><mspace linebreak="newline"/><mo>=</mo><mo> </mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>(</mo><mi mathvariant="normal">t</mi><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>sin</mi><mn>2</mn><mi mathvariant="normal">t</mi><mo>)</mo><msubsup><mo>|</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup><mo> </mo><mo>=</mo><mo> </mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac><mo>.</mo></math></p> <p>c) Đặt u = 1 + xe<sup>x</sup> => du = (e<sup>x</sup> + xe<sup>x</sup>)dx = e<sup>x</sup>(1+x)dx</p> <p>Đổi cận:</p> <table style="border-collapse: collapse; width: 21.4426%; height: 57px;" border="1"> <tbody> <tr> <td style="width: 21.2329%;">x</td> <td style="width: 78.7671%;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mn>1</mn></math></td> </tr> <tr> <td style="width: 21.2329%;">u</td> <td style="width: 78.7671%;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mn>1</mn><mo>+</mo><mi>e</mi></math></td> </tr> </tbody> </table> <p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mfrac><mrow><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup><mo>(</mo><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi><mo>)</mo></mrow><mrow><mn>1</mn><mo>+</mo><msup><mi>xe</mi><mi mathvariant="normal">x</mi></msup></mrow></mfrac><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><msubsup><mo>∫</mo><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mi mathvariant="normal">e</mi></mrow></msubsup><mfrac><mi>du</mi><mi mathvariant="normal">u</mi></mfrac><mo>=</mo><mi>ln</mi><mfenced open="|" close="|"><mi mathvariant="normal">u</mi></mfenced><msubsup><mo>|</mo><mn>1</mn><mrow><mn>1</mn><mo>+</mo><mi mathvariant="normal">e</mi></mrow></msubsup><mo> </mo><mo>=</mo><mo> </mo><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi mathvariant="normal">e</mi><mo>)</mo></math></p> <p>d) Đặt x = a.sint t => dx = a.costdt</p> <p>Đổi cận:</p> <table style="border-collapse: collapse; width: 26.436%; height: 50px;" border="1"> <tbody> <tr> <td style="width: 16.4827%;">x</td> <td style="width: 83.5211%;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfrac><mi>a</mi><mn>2</mn></mfrac></math></td> </tr> <tr> <td style="width: 16.4827%;">t</td> <td style="width: 83.5211%;"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mo> </mo><mfrac><mi>π</mi><mn>6</mn></mfrac></math></td> </tr> </tbody> </table> <p><math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mfrac><mi mathvariant="normal">a</mi><mn>2</mn></mfrac></msubsup><mfrac><mn>1</mn><msqrt><msup><mi mathvariant="normal">a</mi><mn>2</mn></msup><mo>-</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></msqrt></mfrac><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><msubsup><mo>∫</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></msubsup><mfrac><mi>acostdt</mi><msqrt><msup><mi mathvariant="normal">a</mi><mn>2</mn></msup><mo>(</mo><mn>1</mn><mo>-</mo><msup><mi>sin</mi><mn>2</mn></msup><mi mathvariant="normal">t</mi><mo>)</mo></msqrt></mfrac><mi>dt</mi><mo> </mo><mo>=</mo><mo> </mo><msubsup><mo>∫</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></msubsup><mi>dt</mi><mo> </mo><mo>=</mo><mo> </mo><mi mathvariant="normal">t</mi><msubsup><mo>|</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></msubsup><mo> </mo><mo>=</mo><mo> </mo><mfrac><mi mathvariant="normal">π</mi><mn>6</mn></mfrac></math>.</p>