Bài 1: Nguyên hàm
Hướng dẫn giải Bài 4 (Trang 101 SGK Toán Giải tích 12)
<p><strong>Bài 4 (Trang 101 SGK Toán Giải tích 12):</strong></p>
<p>Sử dụng phương pháp tính nguyên hàm từng phần, hãy tính:</p>
<p>a)<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mi>xln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi></mrow></mfenced><mi>dx</mi></math> ;</p>
<p>b) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfenced><mrow><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></mrow></mfenced><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup><mi>dx</mi></math> ;</p>
<p>c) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mi>xsinx</mi><mfenced><mrow><mn>2</mn><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mi>dx</mi></math> ;</p>
<p>d) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">x</mi></mrow></mfenced><mi>cosxdx</mi></math> .</p>
<p><strong><em>Hướng dẫn giải:</em></strong></p>
<p>a) Đặt: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>u</mi><mo>=</mo><mi>ln</mi><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mi>d</mi><mi>v</mi><mo>=</mo><mi>x</mi><mi>d</mi><mi>x</mi></mtd></mtr></mtable></mfenced><mo>⇒</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>d</mi><mi>u</mi><mo>=</mo><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><mn>1</mn><mo>+</mo><mi>x</mi></mrow></mfrac></mtd></mtr><mtr><mtd><mi>v</mi><mo>=</mo><mfrac><msup><mi>x</mi><mn>2</mn></msup><mn>2</mn></mfrac></mtd></mtr></mtable></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mi>xln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi></mrow></mfenced><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><mfrac><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi></mrow></mfenced><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>∫</mo><mfrac><mrow><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mi>dx</mi></mrow><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo> </mo><mo>=</mo><mo> </mo><mfrac><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi></mrow></mfenced><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>∫</mo><mfenced><mrow><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn><mo>+</mo><mfrac><mn>1</mn><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></mfrac></mrow></mfenced><mi>dx</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>=</mo><mo> </mo><mfrac><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi></mrow></mfenced><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mfrac><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>-</mo><mi mathvariant="normal">x</mi><mo>+</mo><mi>ln</mi><mfenced open="|" close="|"><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mrow></mfenced><mo>+</mo><mi mathvariant="normal">C</mi><mo> </mo><mo>=</mo><mo> </mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced><mi>ln</mi><mfenced><mrow><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi></mrow></mfenced><mo>-</mo><mfrac><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mn>4</mn></mfrac><mo>+</mo><mfrac><mi mathvariant="normal">x</mi><mn>2</mn></mfrac><mo>+</mo><mi mathvariant="normal">C</mi></math></p>
<p>b)</p>
<p>Đặt: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><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><mi mathvariant="normal">x</mi></msup><mi>dx</mi></mtd></mtr></mtable></mfenced><mo>⇒</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>du</mi><mo>=</mo><mfenced><mrow><mn>2</mn><mi mathvariant="normal">x</mi><mo>+</mo><mn>2</mn></mrow></mfenced><mi>dx</mi></mtd></mtr><mtr><mtd><mi mathvariant="normal">v</mi><mo>=</mo><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup></mtd></mtr></mtable></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfenced><mrow><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></mrow></mfenced><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup><mi>dx</mi><mo>=</mo><mfenced><mrow><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></mrow></mfenced><mo>.</mo><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup><mo>-</mo><mn>2</mn><mo>∫</mo><mfenced><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>.</mo><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup><mi>dx</mi></math></p>
<p>Đặt: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><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><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup><mi>dx</mi></mtd></mtr></mtable></mfenced><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><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup></mtd></mtr></mtable></mfenced></math></p>
<p>Suy ra <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfenced><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><mfenced><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup><mo>-</mo><mo>∫</mo><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><msup><mi>xe</mi><mi mathvariant="normal">x</mi></msup><mo>+</mo><mi mathvariant="normal">C</mi></math></p>
<p>Vậy <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfenced><mrow><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></mrow></mfenced><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup><mo> </mo><mo>=</mo><mo> </mo><mfenced><mrow><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></mrow></mfenced><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup><mo>-</mo><mn>2</mn><msup><mi>xe</mi><mi mathvariant="normal">x</mi></msup><mo>+</mo><mi mathvariant="normal">C</mi><mo> </mo><mo>=</mo><mo> </mo><mfenced><mrow><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfenced><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup><mo>+</mo><mi mathvariant="normal">C</mi></math></p>
<p>c)</p>
<p>Đặt: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>u</mi><mo>=</mo><mi>x</mi></mtd></mtr><mtr><mtd><mi>d</mi><mi>v</mi><mo>=</mo><mi>sin</mi><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mi>d</mi><mi>x</mi></mtd></mtr></mtable></mfenced><mo>⇒</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>d</mi><mi>u</mi><mo>=</mo><mi>d</mi><mi>x</mi></mtd></mtr><mtr><mtd><mi>v</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>cos</mi><mfenced><mrow><mn>2</mn><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mtd></mtr></mtable></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mi>xsin</mi><mfenced><mrow><mn>2</mn><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><mo>-</mo><mi mathvariant="normal">x</mi></mrow><mn>2</mn></mfrac><mi>cos</mi><mfenced><mrow><mn>2</mn><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>∫</mo><mi>cos</mi><mfenced><mrow><mn>2</mn><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><mfrac><mrow><mo>-</mo><mi mathvariant="normal">x</mi></mrow><mn>2</mn></mfrac><mi>cos</mi><mfenced><mrow><mn>2</mn><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>sin</mi><mfenced><mrow><mn>2</mn><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mi mathvariant="normal">C</mi></math></p>
<p>d) Đặt: <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi mathvariant="normal">u</mi><mo>=</mo><mn>1</mn><mo>-</mo><mi mathvariant="normal">x</mi></mtd></mtr><mtr><mtd><mi>dv</mi><mo>=</mo><mi>cosdx</mi></mtd></mtr></mtable></mfenced><mo>⇒</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>du</mi><mo>=</mo><mo>-</mo><mi>dx</mi></mtd></mtr><mtr><mtd><mi mathvariant="normal">v</mi><mo>=</mo><mi>sinx</mi></mtd></mtr></mtable></mfenced></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">x</mi></mrow></mfenced><mi>cosdx</mi><mo>=</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">x</mi></mrow></mfenced><mi>sinx</mi><mo>+</mo><mo>∫</mo><mi>sinxdx</mi><mo>=</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">x</mi></mrow></mfenced><mi>sinx</mi><mo>-</mo><mi>cosx</mi><mo>+</mo><mi mathvariant="normal">C</mi></math></p>
Hướng dẫn Giải Bài 4 (Trang 101, SGK Toán Giải Tích 12)
GV: 
GV colearn
GV colearn
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Hướng dẫn Giải Bài 4 (Trang 101, SGK Toán Giải Tích 12)
GV: GV colearn
GV colearn