Hướng dẫn giải Bài 5 (Trang 113 SGK Toán Giải tích 12)

Bài 5 (Trang 113 SGK Toán Giải tích 12): Tính các tích phân sau: a) <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mn>3</mn><mi mathvariant="normal">x</mi><mo>)</mo></mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mi>dx</mi></math> ; b)<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></msubsup><mfrac><mrow><msup><mi mathvariant="normal">x</mi><mn>3</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfrac><mi>dx</mi></math> ; c) <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>1</mn><mn>2</mn></msubsup><mfrac><mrow><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi><mo>)</mo></mrow><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mfrac><mi>dx</mi></math> . Hướng dẫn giải: a)<math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mn>3</mn><mi mathvariant="normal">x</mi><mo>)</mo></mrow><mfrac><mn>3</mn><mn>2</mn></mfrac></msup><mi>dx</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>.</mo><mfrac><mn>2</mn><mn>5</mn></mfrac><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mn>3</mn><mi mathvariant="normal">x</mi><mo>)</mo></mrow><mfrac><mn>5</mn><mn>2</mn></mfrac></msup><msubsup><mo>|</mo><mn>0</mn><mn>1</mn></msubsup><mo>=</mo><mfrac><mn>2</mn><mn>15</mn></mfrac><mo>(</mo><msup><mn>4</mn><mfrac><mn>5</mn><mn>2</mn></mfrac></msup><mo>-</mo><mn>1</mn><mo>)</mo><mo>.</mo></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><mi mathvariant="normal">b</mi><mo>)</mo><mo> </mo><msubsup><mo>∫</mo><mn>0</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></msubsup><mfrac><mrow><msup><mi mathvariant="normal">x</mi><mn>3</mn></msup><mo>-</mo><mn>1</mn></mrow><mrow><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>-</mo><mn>1</mn></mrow></mfrac><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><msubsup><mo>∫</mo><mn>0</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></msubsup><mfrac><mrow><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mo>+</mo><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><msubsup><mo>∫</mo><mn>0</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></msubsup><mo>(</mo><mi mathvariant="normal">x</mi><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><mo>(</mo><mfrac><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>+</mo><mi>ln</mi><mfenced open="|" close="|"><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></mfenced><mo>)</mo><mrow><msubsup><mo>|</mo><mn>0</mn><mfrac><mn>1</mn><mn>2</mn></mfrac></msubsup><mo> </mo></mrow><mo> </mo><mo>=</mo><mfrac><mn>1</mn><mn>8</mn></mfrac><mo>+</mo><mi>ln</mi><mfrac><mn>3</mn><mn>2</mn></mfrac></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><mfrac><mn>1</mn><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mfrac><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><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><mfrac><mn>1</mn><mi mathvariant="normal">x</mi></mfrac></mtd></mtr></mtable></mfenced></mrow></mfenced></math> <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>1</mn><mn>2</mn></msubsup><mfrac><mrow><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi><mo>)</mo></mrow><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup></mfrac><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mfrac><mrow><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi><mo>)</mo></mrow><mi mathvariant="normal">x</mi></mfrac><msubsup><mo>|</mo><mn>1</mn><mn>2</mn></msubsup><mo> </mo><mo>+</mo><msubsup><mo>∫</mo><mn>1</mn><mn>2</mn></msubsup><mfrac><mi>dx</mi><mrow><mi mathvariant="normal">x</mi><mo>(</mo><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></mfrac><mo> </mo><mo>=</mo><mo> </mo><mo>-</mo><mfrac><mrow><mi>ln</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi mathvariant="normal">x</mi><mo>)</mo></mrow><mi mathvariant="normal">x</mi></mfrac><msubsup><mo>|</mo><mn>1</mn><mn>2</mn></msubsup><mo> </mo><mo>+</mo><mo> </mo><mi>ln</mi><mfrac><mi mathvariant="normal">x</mi><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><msubsup><mo>|</mo><mn>1</mn><mn>2</mn></msubsup><mo> </mo><mo>=</mo><mo> </mo><mn>3</mn><mi>ln</mi><mfrac><mrow><mn>2</mn><msqrt><mn>3</mn></msqrt></mrow><mn>3</mn></mfrac></math>