Question

В 4 и 6 примере, интеграл растянут на всю дробь.

Answer 1

4

$\int\limits^{ 1 } _ {0} \frac{dx}{ 9 + {x}^{2} } = \int\limits^{ 1 } _ {0} \frac{dx}{ {3}^{2} + {x}^{2} } = \frac{1}{3}arctg( \frac{x}{3}) | ^{ 1 } _ {0} = \\ = \frac{1}{3} (arctg( \frac{1}{3}) - arctg(0)) = \frac{1}{3} arctg (\frac{1}{3} )$

5

$\int\limits^{ 4 } _ {2 \sqrt{2} }x \sqrt{ {x}^{2} - 7} dx \\ \\ {x}^{2} - 7 = t {}^{2} \\ x {}^{2} = t {}^{2} + 7 \\ x = \sqrt{ {t}^{2} + 7 } \\ dx = \frac{2tdt}{2 \sqrt{ {t}^{2} + 7 } } = \frac{tdt}{ \sqrt{ {t}^{2} + 7} } \\ t1 = \sqrt{ {4}^{2} - 7} = 3 \\ t2 = \sqrt{8 - 7} = 1 \\ \\ \\ \int\limits^{ 3 } _ {1} \sqrt{ {t}^{2} + 7} \times t \times \frac{tdt}{ \sqrt{ {t}^{2} + 7} } = \\ = \int\limits^{ 3 } _ {1} {t}^{2}dt = \ \frac{ {t}^{3} }{3} | ^{3 } _ {1} = 9 - \frac{1}{3} = 8 \frac{2}{3}$

6

$\int\limits^{ e } _ {1} \frac{ ln(x) }{x} dx \\ \\ ln(x) = t \\ \frac{dx}{x} = dt \\ t1 = ln(e) = 1 \\ t2 = ln(1) = 0 \\ \\ \int\limits^{ 1 } _ {0}tdt = \frac{ {t}^{2} }{2} | ^{ 1 } _ {0} = \frac{1}{2} - 0 = 0.5$