Hướng dẫn giải Bài 4 (Trang 113 SGK Toán Giải tích 12)

Bài 4 (Trang 113 SGK Toán Giải tích 12): Sử dụng phương pháp tích phân từng phần, tính các tích phân: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><msubsup><mo>∫</mo><mn>0</mn><mfrac><mi>π</mi><mn>2</mn></mfrac></msubsup><mo>(</mo><mi>x</mi><mo>+</mo><mn>1</mn><mo>)</mo><mi>sin</mi><mi>x</mi><mi>d</mi><mi>x</mi><mspace linebreak="newline"/><mi>b</mi><mo>)</mo><mo> </mo><msubsup><mo>∫</mo><mn>1</mn><mi>e</mi></msubsup><msup><mi>x</mi><mn>2</mn></msup><mi>ln</mi><mi>x</mi><mi>d</mi><mi>x</mi><mspace linebreak="newline"/><mi>c</mi><mo>)</mo><mo> </mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo><mi>d</mi><mi>x</mi><mspace linebreak="newline"/><mi>d</mi><mo>)</mo><mo> </mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mo>(</mo><msup><mi>x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>1</mn><mo>)</mo><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup><mi>d</mi><mi>x</mi><mo>.</mo></math> Hướng dẫn giải: a) Đặt <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">u</mi><mo>=</mo><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mtd></mtr><mtr><mtd><mi>dv</mi><mo>=</mo><mi>sinxdx</mi></mtd></mtr></mtable><mo>⇒</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>du</mi><mo>=</mo><mi>dx</mi></mtd></mtr><mtr><mtd><mi mathvariant="normal">v</mi><mo>=</mo><mo>-</mo><mi>cosx</mi></mtd></mtr></mtable></mfenced></mrow></mfenced></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup><mo>(</mo><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn><mo>)</mo><mi>sinxdx</mi><mo> </mo><mo>=</mo><mo>-</mo><mo>(</mo><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn><mo>)</mo><mi>cosx</mi><msubsup><mo>|</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup><mo> </mo><mo>+</mo><msubsup><mo>∫</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup><mi>cosxdx</mi><mo> </mo><mo>=</mo><mo>-</mo><mo>(</mo><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn><mo>)</mo><mi>cosx</mi><msubsup><mo>|</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup><mo> </mo><mo>+</mo><mi>sinx</mi><msubsup><mo>|</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup><mo> </mo><mo>=</mo><mo> </mo><mn>2</mn></math> b) Đặt <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">u</mi><mo>=</mo><mi>lnx</mi></mtd></mtr><mtr><mtd><mi>dv</mi><mo>=</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mi>dx</mi></mtd></mtr></mtable><mo>⇒</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>du</mi><mo>=</mo><mfrac><mi>dx</mi><mi mathvariant="normal">x</mi></mfrac></mtd></mtr><mtr><mtd><mi mathvariant="normal">v</mi><mo>=</mo><mfrac><msup><mi mathvariant="normal">x</mi><mn>3</mn></msup><mn>3</mn></mfrac></mtd></mtr></mtable></mfenced></mrow></mfenced></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>1</mn><mi mathvariant="normal">e</mi></msubsup><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mi>lnxdx</mi><mo> </mo><mo>=</mo><mo> </mo><mfrac><msup><mi mathvariant="normal">x</mi><mn>3</mn></msup><mn>3</mn></mfrac><mi>lnx</mi><msubsup><mo>|</mo><mn>1</mn><mi mathvariant="normal">e</mi></msubsup><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><msubsup><mo>∫</mo><mn>1</mn><mi mathvariant="normal">e</mi></msubsup><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><mfrac><msup><mi mathvariant="normal">x</mi><mn>3</mn></msup><mn>3</mn></mfrac><mi>lnx</mi><msubsup><mo>|</mo><mn>1</mn><mi mathvariant="normal">e</mi></msubsup><mo>-</mo><mfrac><msup><mi mathvariant="normal">x</mi><mn>3</mn></msup><mn>9</mn></mfrac><msubsup><mo>|</mo><mn>1</mn><mi mathvariant="normal">e</mi></msubsup><mo> </mo><mo> </mo><mo>=</mo><mo> </mo><mfrac><mn>1</mn><mn>9</mn></mfrac><mo>(</mo><mn>2</mn><msup><mi mathvariant="normal">e</mi><mn>3</mn></msup><mo>+</mo><mn>1</mn><mo>)</mo></math> c) Đặt <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">u</mi><mo>=</mo><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi><mo>)</mo></mtd></mtr><mtr><mtd><mi>dv</mi><mo>=</mo><mi>dx</mi></mtd></mtr></mtable><mo> </mo><mo>⇒</mo><mo> </mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>du</mi><mo> </mo><mo>=</mo><mo> </mo><mfrac><mi>dx</mi><mrow><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mi mathvariant="normal">v</mi><mo> </mo><mo>=</mo><mo> </mo><mi mathvariant="normal">x</mi></mtd></mtr></mtable></mfenced></mrow></mfenced></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi><mo>)</mo><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><mi>xln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi><mo>)</mo><msubsup><mo>|</mo><mn>0</mn><mn>1</mn></msubsup><mo>-</mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mfrac><mi mathvariant="normal">x</mi><mrow><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi></mrow></mfrac><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><mi>xln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi><mo>)</mo><msubsup><mo>|</mo><mn>0</mn><mn>1</mn></msubsup><mo>-</mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mo>(</mo><mn>1</mn><mo>-</mo><mfrac><mi mathvariant="normal">x</mi><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>)</mo><mi>dx</mi></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo> </mo><mi>xln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi><mo>)</mo><msubsup><mo>|</mo><mn>0</mn><mn>1</mn></msubsup><mo> </mo><mo>-</mo><mo> </mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mo>(</mo><mn>1</mn><mo>-</mo><mfrac><mn>1</mn><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>)</mo><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><mi>ln</mi><mn>2</mn><mo>-</mo><mo>(</mo><mi mathvariant="normal">x</mi><mo>-</mo><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi><mo>)</mo><msubsup><mo>|</mo><mn>0</mn><mn>1</mn></msubsup><mo> </mo><mo> </mo><mo>=</mo><mo> </mo><mn>2</mn><mi>ln</mi><mn>2</mn><mo>-</mo><mn>1</mn></math> d) Đặt <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">u</mi><mo>=</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn></mtd></mtr><mtr><mtd><mi>dv</mi><mo>=</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>-</mo><mi mathvariant="normal">x</mi></mrow></msup><mi>dx</mi></mtd></mtr></mtable><mo>⇒</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>du</mi><mo>=</mo><mo>(</mo><mn>2</mn><mi mathvariant="normal">x</mi><mo>-</mo><mn>2</mn><mo>)</mo><mi>dx</mi></mtd></mtr><mtr><mtd><mi mathvariant="normal">v</mi><mo>=</mo><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>-</mo><mi mathvariant="normal">x</mi></mrow></msup></mtd></mtr></mtable></mfenced></mrow></mfenced></math>    <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mo>(</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn><mo>)</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>-</mo><mi mathvariant="normal">x</mi></mrow></msup><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mo>(</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn><mo>)</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>-</mo><mi mathvariant="normal">x</mi></mrow></msup><msubsup><mo>|</mo><mn>0</mn><mn>1</mn></msubsup><mo>+</mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mo>(</mo><mn>2</mn><mi mathvariant="normal">x</mi><mo>-</mo><mn>2</mn><mo>)</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>-</mo><mi mathvariant="normal">x</mi></mrow></msup><mi>dx</mi></math>     Đặt <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mrow><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">u</mi><mo>=</mo><mn>2</mn><mi mathvariant="normal">x</mi><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mi>dv</mi><mo>=</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>-</mo><mi mathvariant="normal">x</mi></mrow></msup><mi>dx</mi></mtd></mtr></mtable><mo>⇒</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>du</mi><mo>=</mo><mn>2</mn><mi>dx</mi></mtd></mtr><mtr><mtd><mi mathvariant="normal">v</mi><mo>=</mo><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>-</mo><mi mathvariant="normal">x</mi></mrow></msup></mtd></mtr></mtable></mfenced></mrow></mfenced></math>   <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mo>(</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>2</mn><mo>)</mo><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup><mi>d</mi><mi>x</mi><mo>=</mo><mo>-</mo><mo>(</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>2</mn><mo>)</mo><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup><msubsup><mo>|</mo><mn>0</mn><mn>1</mn></msubsup><mo>+</mo><mn>2</mn><msubsup><mo>∫</mo><mrow/><mrow/></msubsup><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup><mi>d</mi><mi>x</mi><mo>=</mo><mo>-</mo><mo>(</mo><mn>2</mn><mi>x</mi><mo>-</mo><mn>2</mn><mo>)</mo><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup><msubsup><mo>|</mo><mn>0</mn><mn>1</mn></msubsup><mo>-</mo><mn>2</mn><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup><msubsup><mo>|</mo><mn>0</mn><mn>1</mn></msubsup></math> Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mo>(</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn><mo>)</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>-</mo><mi mathvariant="normal">x</mi></mrow></msup><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mo>(</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mn>2</mn><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn><mo>)</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>-</mo><mi mathvariant="normal">x</mi></mrow></msup><msubsup><mo>|</mo><mn>0</mn><mn>1</mn></msubsup><mo> </mo><mo>-</mo><mo>(</mo><mn>2</mn><mi mathvariant="normal">x</mi><mo>-</mo><mn>2</mn><mo>)</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>-</mo><mi mathvariant="normal">x</mi></mrow></msup><msubsup><mo>|</mo><mn>0</mn><mn>1</mn></msubsup><mo> </mo><mo>-</mo><mn>2</mn><msup><mi mathvariant="normal">e</mi><mrow><mo>-</mo><mi mathvariant="normal">x</mi></mrow></msup><msubsup><mo>|</mo><mn>0</mn><mn>1</mn></msubsup><mo> </mo><mo>=</mo><mo>-</mo><mn>1</mn></math>