Bài 1: Nguyên hàm
Hướng dẫn giải Bài 2 (Trang 100 SGK Toán Giải tích 12)
<p><strong>Bài 2 (Trang 100 SGK Toán Giải tích 12):</strong></p>
<p>Tìm nguyên hàm của các hàm số sau:</p>
<p>a) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">f</mi><mfenced><mi mathvariant="normal">x</mi></mfenced><mo>=</mo><mfrac><mrow><mi mathvariant="normal">x</mi><mo>+</mo><msqrt><mi mathvariant="normal">x</mi></msqrt><mo>+</mo><mn>1</mn></mrow><mroot><mi mathvariant="normal">x</mi><mn>3</mn></mroot></mfrac></math>;</p>
<p>b) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">f</mi><mfenced><mi mathvariant="normal">x</mi></mfenced><mo>=</mo><mo> </mo><mfrac><mrow><msup><mn>2</mn><mi mathvariant="normal">x</mi></msup><mo>-</mo><mn>1</mn></mrow><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup></mfrac></math>;</p>
<p>c) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">f</mi><mfenced><mi mathvariant="normal">x</mi></mfenced><mo>=</mo><mfrac><mn>1</mn><mrow><msup><mi>sin</mi><mn>2</mn></msup><mi mathvariant="normal">x</mi><mo>.</mo><msup><mi>cos</mi><mn>2</mn></msup><mi mathvariant="normal">x</mi></mrow></mfrac></math>;</p>
<p>d) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">f</mi><mo>(</mo><mi mathvariant="normal">x</mi><mo>)</mo><mo> </mo><mo>=</mo><mo> </mo><mi>sin</mi><mn>5</mn><mi mathvariant="normal">x</mi><mo>.</mo><mi>cos</mi><mn>3</mn><mi mathvariant="normal">x</mi></math>;</p>
<p>e) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">f</mi><mo>(</mo><mi mathvariant="normal">x</mi><mo>)</mo><mo>=</mo><msup><mi>tan</mi><mn>2</mn></msup><mi mathvariant="normal">x</mi></math>;</p>
<p>g) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">f</mi><mo>(</mo><mi mathvariant="normal">x</mi><mo>)</mo><mo>=</mo><mo> </mo><msup><mi mathvariant="normal">e</mi><mrow><mn>3</mn><mo>-</mo><mn>2</mn><mi mathvariant="normal">x</mi></mrow></msup></math>;</p>
<p>h) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">f</mi><mfenced><mi mathvariant="normal">x</mi></mfenced><mo>=</mo><mo> </mo><mfrac><mn>1</mn><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi><mo>)</mo><mo>(</mo><mn>1</mn><mo>-</mo><mn>2</mn><mi mathvariant="normal">x</mi><mo>)</mo></mrow></mfrac></math>;</p>
<p><em><strong>Hướng dẫn giải:</strong></em></p>
<p>a) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">f</mi><mfenced><mi mathvariant="normal">x</mi></mfenced><mo> </mo><mo>=</mo><mo> </mo><mfrac><mi mathvariant="normal">x</mi><msup><mi mathvariant="normal">x</mi><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle></msup></mfrac><mo> </mo><mo>+</mo><mo> </mo><mfrac><msup><mi mathvariant="normal">x</mi><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></msup><msup><mi mathvariant="normal">x</mi><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle></msup></mfrac><mo> </mo><mo>+</mo><mo> </mo><mfrac><mn>1</mn><msup><mi mathvariant="normal">x</mi><mstyle displaystyle="true"><mfrac><mn>1</mn><mn>3</mn></mfrac></mstyle></msup></mfrac><mo> </mo><mo>=</mo><mo> </mo><msup><mi mathvariant="normal">x</mi><mfrac><mn>2</mn><mn>3</mn></mfrac></msup><mo> </mo><mo>+</mo><mo> </mo><msup><mi mathvariant="normal">x</mi><mfrac><mn>1</mn><mn>6</mn></mfrac></msup><mo> </mo><mo>+</mo><mo> </mo><msup><mi mathvariant="normal">x</mi><mrow><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow></msup></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>∫</mo><mi mathvariant="normal">f</mi><mfenced><mi mathvariant="normal">x</mi></mfenced><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><mfrac><mn>3</mn><mn>5</mn></mfrac><msup><mi mathvariant="normal">x</mi><mfrac><mn>5</mn><mn>3</mn></mfrac></msup><mo>+</mo><mo> </mo><mfrac><mn>6</mn><mn>7</mn></mfrac><msup><mi mathvariant="normal">x</mi><mfrac><mn>7</mn><mn>6</mn></mfrac></msup><mo>+</mo><mo> </mo><mfrac><mn>3</mn><mn>2</mn></mfrac><msup><mi mathvariant="normal">x</mi><mfrac><mn>2</mn><mn>3</mn></mfrac></msup><mo> </mo><mo>+</mo><mo> </mo><mi mathvariant="normal">C</mi></math></p>
<p>b) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">f</mi><mfenced><mi mathvariant="normal">x</mi></mfenced><mo>=</mo><mo> </mo><msup><mrow><mo>(</mo><mfrac><mn>2</mn><mi mathvariant="normal">e</mi></mfrac><mo>)</mo></mrow><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></math></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo>⇒</mo><mo>∫</mo><mi mathvariant="normal">f</mi><mfenced><mi mathvariant="normal">x</mi></mfenced><mi>dx</mi><mo>=</mo><mo> </mo><mfrac><msup><mfenced><mstyle displaystyle="true"><mfrac><mn>2</mn><mi mathvariant="normal">e</mi></mfrac></mstyle></mfenced><mi mathvariant="normal">x</mi></msup><mrow><mi>ln</mi><mstyle displaystyle="true"><mfrac><mn>2</mn><mi mathvariant="normal">e</mi></mfrac></mstyle></mrow></mfrac><mo>+</mo><mo> </mo><msup><mi mathvariant="normal">e</mi><mrow><mo>-</mo><mi mathvariant="normal">x</mi></mrow></msup><mo>=</mo><mfrac><msup><mn>2</mn><mi mathvariant="normal">x</mi></msup><mrow><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup><mfenced><mrow><mi>ln</mi><mn>2</mn><mo>-</mo><mn>1</mn></mrow></mfenced></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup></mfrac><mo>=</mo><mfrac><mrow><msup><mn>2</mn><mi mathvariant="normal">x</mi></msup><mo>+</mo><mi>ln</mi><mn>2</mn><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup><mo>(</mo><mi>ln</mi><mn>2</mn><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mfrac></math></p>
<p>c)<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><mfrac><mi>dx</mi><mrow><msup><mi>sin</mi><mn>2</mn></msup><msup><mi>xcos</mi><mn>2</mn></msup><mi mathvariant="normal">x</mi></mrow></mfrac><mo>=</mo><mo>∫</mo><mfenced><mrow><mfrac><mn>1</mn><mrow><msup><mi>sin</mi><mn>2</mn></msup><mi mathvariant="normal">x</mi></mrow></mfrac><mo>+</mo><mo> </mo><mfrac><mn>1</mn><mrow><msup><mi>cos</mi><mn>2</mn></msup><mi mathvariant="normal">x</mi></mrow></mfrac></mrow></mfenced><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><mi>tanx</mi><mo> </mo><mo>-</mo><mo> </mo><mi>cotx</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"><mo>∫</mo><mi>sin</mi><mn>5</mn><mi>xcos</mi><mn>3</mn><mi>xdx</mi><mo> </mo><mo>=</mo><mo> </mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>∫</mo><mfenced><mrow><mi>sin</mi><mn>8</mn><mi mathvariant="normal">x</mi><mo> </mo><mo>+</mo><mo> </mo><mi>sin</mi><mn>2</mn><mi mathvariant="normal">x</mi></mrow></mfenced><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mfenced><mrow><mfrac><mn>1</mn><mn>8</mn></mfrac><mi>cos</mi><mn>8</mn><mi mathvariant="normal">x</mi><mo> </mo><mo>+</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>cos</mi><mn>2</mn><mi mathvariant="normal">x</mi></mrow></mfenced><mo> </mo><mo>+</mo><mo> </mo><mi mathvariant="normal">C</mi><mo> </mo><mspace linebreak="newline"/><mo>=</mo><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mfenced><mrow><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>cos</mi><mn>8</mn><mi mathvariant="normal">x</mi><mo> </mo><mo>+</mo><mo> </mo><mi>cos</mi><mn>2</mn><mi mathvariant="normal">x</mi></mrow></mfenced><mo>+</mo><mo> </mo><mi mathvariant="normal">C</mi></math></p>
<p>e)<math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><msup><mi>tan</mi><mn>2</mn></msup><mi>xdx</mi><mo> </mo><mo>=</mo><mo> </mo><mo>∫</mo><mfenced><mrow><mfrac><mn>1</mn><mrow><msup><mi>cos</mi><mn>2</mn></msup><mi mathvariant="normal">x</mi></mrow></mfrac><mo>-</mo><mn>1</mn></mrow></mfenced><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><mi>tanx</mi><mo>-</mo><mi mathvariant="normal">x</mi><mo> </mo><mo>+</mo><mo> </mo><mi mathvariant="normal">C</mi></math></p>
<p>g) <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>∫</mo><msup><mi mathvariant="normal">e</mi><mrow><mn>3</mn><mo>-</mo><mn>2</mn><mi mathvariant="normal">x</mi></mrow></msup><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msup><mi mathvariant="normal">e</mi><mrow><mn>3</mn><mo>-</mo><mn>2</mn><mi mathvariant="normal">x</mi></mrow></msup><mo>+</mo><mo> </mo><mi mathvariant="normal">C</mi></math></p>
<p>h) <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">f</mi><mfenced><mi mathvariant="normal">x</mi></mfenced><mo>=</mo><mfrac><mn>2</mn><mrow><mfenced><mrow><mn>2</mn><mo>+</mo><mn>2</mn><mi mathvariant="normal">x</mi></mrow></mfenced><mfenced><mrow><mn>1</mn><mo>-</mo><mn>2</mn><mi mathvariant="normal">x</mi></mrow></mfenced></mrow></mfrac><mo>=</mo><mo> </mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mfenced><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo>+</mo><mn>2</mn><mi mathvariant="normal">x</mi></mrow></mfrac><mo>+</mo><mfrac><mn>1</mn><mrow><mn>1</mn><mo>-</mo><mn>2</mn><mi mathvariant="normal">x</mi></mrow></mfrac></mrow></mfenced><mo> </mo><mo>=</mo><mo> </mo><mfrac><mn>2</mn><mn>3</mn></mfrac><mfenced><mrow><mfrac><mn>1</mn><mrow><mn>2</mn><mo>(</mo><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mfrac><mo>-</mo><mfrac><mn>1</mn><mrow><mn>2</mn><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></mrow></mfenced><mspace linebreak="newline"/><mo>⇒</mo><mo>∫</mo><mi mathvariant="normal">f</mi><mfenced><mi mathvariant="normal">x</mi></mfenced><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mfenced open="|" close="|"><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>-</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mfenced open="|" close="|"><mrow><mn>2</mn><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn></mrow></mfenced><mo>+</mo><mi mathvariant="normal">C</mi><mo>=</mo><mfrac><mn>1</mn><mn>3</mn></mfrac><mi>ln</mi><mfenced open="|" close="|"><mfrac><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn></mrow></mfrac></mfenced><mo>+</mo><mi mathvariant="normal">C</mi></math></p>
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