Hướng dẫn giải Bài 2 (Trang 112 SGK Toán Giải tích 12)

Bài 2 (Trang 112 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>2</mn></msubsup><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">x</mi></mrow></mfenced><mi>dx</mi></math>; b) <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><msup><mi>sin</mi><mn>2</mn></msup><mi>xdx</mi></math>; c) <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mrow><mi>ln</mi><mn>2</mn></mrow></msubsup><mfrac><mrow><msup><mi mathvariant="normal">e</mi><mrow><mn>2</mn><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><mn>1</mn></mrow><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup></mfrac><mi>dx</mi></math>; d) <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mi mathvariant="normal">π</mi></msubsup><mi>sin</mi><mn>2</mn><msup><mi>xcos</mi><mn>2</mn></msup><mi>xdx</mi></math>. Hướng dẫn giải: a) Ta có: <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mn>2</mn></msubsup><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">x</mi></mrow></mfenced><mi>dx</mi><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">x</mi></mrow></mfenced><mi>dx</mi><mo> </mo><mo>+</mo><mo> </mo><msubsup><mo>∫</mo><mn>1</mn><mn>2</mn></msubsup><mfenced open="|" close="|"><mrow><mn>1</mn><mo>-</mo><mi mathvariant="normal">x</mi></mrow></mfenced><mi>dx</mi><mo> </mo><mo>=</mo><msubsup><mo>∫</mo><mn>0</mn><mn>1</mn></msubsup><mo>(</mo><mn>1</mn><mo>-</mo><mi mathvariant="normal">x</mi><mo>)</mo><mi>dx</mi><mo>+</mo><msubsup><mo>∫</mo><mn>1</mn><mn>2</mn></msubsup><mo>(</mo><mi mathvariant="normal">x</mi><mo>-</mo><mn>1</mn><mo>)</mo><mi>dx</mi><mo>=</mo><mo>(</mo><mi mathvariant="normal">x</mi><mo>-</mo><mfrac><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>)</mo><msubsup><mo>|</mo><mn>0</mn><mn>1</mn></msubsup><mo> </mo><mo>+</mo><mo>(</mo><mfrac><msup><mi mathvariant="normal">x</mi><mn>2</mn></msup><mn>2</mn></mfrac><mo>-</mo><mi mathvariant="normal">x</mi><mo>)</mo><msubsup><mo>|</mo><mn>1</mn><mn>2</mn></msubsup><mo>=</mo><mn>1</mn></math> b) <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><msup><mi>sin</mi><mn>2</mn></msup><mi>xdx</mi><mo> </mo><mo>=</mo><mo> </mo><msubsup><mo>∫</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup><mfrac><mrow><mn>1</mn><mo>-</mo><mi>cos</mi><mn>2</mn><mi mathvariant="normal">x</mi></mrow><mn>2</mn></mfrac><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>(</mo><mi mathvariant="normal">x</mi><mo>-</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mi>sin</mi><mn>2</mn><mi mathvariant="normal">x</mi><mo>)</mo><msubsup><mo>|</mo><mn>0</mn><mfrac><mi mathvariant="normal">π</mi><mn>2</mn></mfrac></msubsup><mo> </mo><mo>=</mo><mo> </mo><mfrac><mi mathvariant="normal">π</mi><mn>4</mn></mfrac></math> c) <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mrow><mi>ln</mi><mn>2</mn></mrow></msubsup><mfrac><mrow><msup><mi mathvariant="normal">e</mi><mrow><mn>2</mn><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><mn>1</mn></mrow><msup><mi mathvariant="normal">e</mi><mi mathvariant="normal">x</mi></msup></mfrac><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><msubsup><mo>∫</mo><mn>0</mn><mrow><mi>ln</mi><mn>2</mn></mrow></msubsup><mo>(</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>+</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>-</mo><mi mathvariant="normal">x</mi></mrow></msup><mo>)</mo><mi>dx</mi><mo> </mo><mo>=</mo><mo> </mo><mo>(</mo><msup><mi mathvariant="normal">e</mi><mrow><mi mathvariant="normal">x</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>-</mo><msup><mi mathvariant="normal">e</mi><mrow><mo>-</mo><mi mathvariant="normal">x</mi></mrow></msup><mo>)</mo><msubsup><mo>|</mo><mn>0</mn><mrow><mi>ln</mi><mn>2</mn></mrow></msubsup><mo> </mo><mo>=</mo><mo> </mo><mi mathvariant="normal">e</mi><mo>+</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle></math> d) <math xmlns="http://www.w3.org/1998/Math/MathML"><msubsup><mo>∫</mo><mn>0</mn><mi>π</mi></msubsup><mi>sin</mi><mn>2</mn><msup><mi>xcos</mi><mn>2</mn></msup><mi>xdx</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msubsup><mo>∫</mo><mn>0</mn><mi mathvariant="normal">π</mi></msubsup><mi>sin</mi><mn>2</mn><mi mathvariant="normal">x</mi><mo>(</mo><mn>1</mn><mo>+</mo><mi>cos</mi><mn>2</mn><mi mathvariant="normal">x</mi><mo>)</mo><mi>dx</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msubsup><mo>∫</mo><mn>0</mn><mi mathvariant="normal">π</mi></msubsup><mi>sin</mi><mn>2</mn><mi>xdx</mi><mo> </mo><mo>+</mo><mo> </mo><mfrac><mn>1</mn><mn>4</mn></mfrac><msubsup><mo>∫</mo><mn>0</mn><mi mathvariant="normal">π</mi></msubsup><mi>sin</mi><mn>4</mn><mi>xdx</mi><mo> </mo><mo>=</mo><mo>(</mo><mo>-</mo><mfrac><mn>1</mn><mn>4</mn></mfrac><mi>cos</mi><mn>2</mn><mi mathvariant="normal">x</mi><mo>-</mo><mfrac><mn>1</mn><mn>16</mn></mfrac><mi>cos</mi><mn>4</mn><mi mathvariant="normal">x</mi><mo>)</mo><msubsup><mo>|</mo><mn>0</mn><mi mathvariant="normal">π</mi></msubsup><mo> </mo><mo>=</mo><mo> </mo><mn>0</mn></math>