Bài 1: Nguyên hàm
Hướng dẫn giải Bài 3 (Trang 101 SGK Toán Giải tích 12)
<p><strong>Bài 3 (Trang 101 SGK Toán Giải tích 12):</strong></p>
<p>Sử dụng phương pháp đổi biến số, hãy tính :</p>
<p>a) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi>x</mi></mrow></mfenced><mn>9</mn></msup><mo> </mo><mi>d</mi><mi>x</mi><mo> </mo><mo> </mo><mo> </mo></math>(đặt <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>=</mo><mn>1</mn><mo>-</mo><mi>x</mi></math>) ;</p>
<p>b) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mi>x</mi><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><msup><mi>x</mi><mn>2</mn></msup><mo>)</mo></mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mi>d</mi><mi>x</mi><mo> </mo></math> (đặt u = 1 + x<sup>2</sup>);</p>
<p>c) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><msup><mi>cos</mi><mn>3</mn></msup><mi>x</mi><mi>sin</mi><mi>x</mi><mi>d</mi><mi>x</mi><mo> </mo></math>(đặt t = cos x);</p>
<p>d) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mrow><mi>d</mi><mi>x</mi></mrow><mrow><msup><mi>e</mi><mi>x</mi></msup><mo> </mo><mo>+</mo><msup><mi>e</mi><mrow><mo>-</mo><mi>x</mi></mrow></msup><mo>+</mo><mn>2</mn></mrow></mfrac></math> (đặt u = e<sup>x</sup> + 1).</p>
<p><strong>Hướng dẫn giải:</strong></p>
<p>a) Đặt u = 1 - x => du = -dx => dx = -du</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">x</mi></mrow></mfenced><mn>9</mn></msup><mi>dx</mi><mo> </mo><mo>=</mo><mo>-</mo><mo>∫</mo><msup><mi mathvariant="normal">u</mi><mrow><mn>9</mn><mo> </mo></mrow></msup><mi>du</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mfrac><msup><mi mathvariant="normal">u</mi><mn>10</mn></msup><mn>10</mn></mfrac><mo>+</mo><mi mathvariant="normal">C</mi><mo> </mo><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>10</mn></mfrac><msup><mfenced><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">x</mi></mrow></mfenced><mn>10</mn></msup><mo>+</mo><mi mathvariant="normal">C</mi></math></p>
<p>b) Đặt <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">u</mi><mo>=</mo><mo> </mo><mn>1</mn><mo>+</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>⇒</mo><mi>du</mi><mo>=</mo><mn>2</mn><mi>xdx</mi><mo> </mo><mo>⇒</mo><mi>xdx</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>du</mi></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mi mathvariant="normal">x</mi><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mrow></mfenced><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mi>dx</mi><mo>=</mo><mo> </mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>∫</mo><msup><mi mathvariant="normal">u</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mi>du</mi><mo>=</mo><mo> </mo><mfrac><mn>1</mn><mn>5</mn></mfrac><msup><mi mathvariant="normal">u</mi><mfrac><mn>5</mn><mn>2</mn></mfrac></msup><mo>+</mo><mo> </mo><mi mathvariant="normal">C</mi><mo> </mo><mo>=</mo><mo> </mo><mfrac><mn>1</mn><mn>5</mn></mfrac><msup><mfenced><mrow><mn>1</mn><mo>+</mo><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mrow></mfenced><mfrac><mn>5</mn><mn>2</mn></mfrac></msup><mo> </mo><mo>+</mo><mi mathvariant="normal">C</mi></math></p>
<p>c) Đặt t = cos x => dt = -sinxdx => sinxdx = -dt</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><msup><mi>cos</mi><mn>3</mn></msup><mi>xsinxdx</mi><mo> </mo><mo>=</mo><mo>-</mo><mo>∫</mo><msup><mi mathvariant="normal">t</mi><mn>3</mn></msup><mi>dt</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mfrac><msup><mi mathvariant="normal">t</mi><mn>4</mn></msup><mn>4</mn></mfrac><mo>+</mo><mo> </mo><mi mathvariant="normal">C</mi><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msup><mi>cos</mi><mn>4</mn></msup><mi mathvariant="normal">x</mi><mo> </mo><mo>+</mo><mo> </mo><mi mathvariant="normal">C</mi></math></p>
<p>d) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">I</mi><mo>=</mo><mo>∫</mo><mfrac><mi>dx</mi><mrow><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup><mo>+</mo><mo> </mo><msup><mi mathvariant="normal">e</mi><mrow><mo>-</mo><mi mathvariant="normal">x</mi></mrow></msup><mo>+</mo><mn>2</mn></mrow></mfrac><mo>=</mo><mo> </mo><mo>∫</mo><mfrac><mrow><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup><mi>dx</mi></mrow><mrow><msup><mi mathvariant="normal">e</mi><mrow><mn>2</mn><mi mathvariant="normal">x</mi></mrow></msup><mo>+</mo><mn>2</mn><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup><mo>+</mo><mn>1</mn><mo> </mo></mrow></mfrac><mo>=</mo><mo>∫</mo><mfrac><mrow><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup><mi>dx</mi></mrow><msup><mfenced><mrow><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup><mo>+</mo><mn>1</mn></mrow></mfenced><mn>2</mn></msup></mfrac></math></p>
<p>Đặt t = e<sup>x</sup> + 1 => dt = e<sup>x</sup>dx</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo> </mo><mi mathvariant="normal">I</mi><mo>=</mo><mo> </mo><mo>∫</mo><mfrac><mi>dt</mi><msup><mi mathvariant="normal">t</mi><mn>2</mn></msup></mfrac><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mi mathvariant="normal">t</mi></mfrac><mo>+</mo><mo> </mo><mi mathvariant="normal">C</mi><mo> </mo><mo>=</mo><mfrac><mrow><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup><mo>+</mo><mn>1</mn><mo> </mo></mrow></mfrac><mo>+</mo><mi mathvariant="normal">C</mi></math></p>
Hướng dẫn Giải Bài 3 (Trang 101, SGK Toán Giải Tích 12)
GV: 
GV colearn
GV colearn
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Hướng dẫn Giải Bài 3 (Trang 101, SGK Toán Giải Tích 12)
GV: GV colearn
GV colearn