Question

Помогите, пожалуйста. Найти интегралы.

Answer 1

Ответ:

$1)∫( \frac{1}{3} {x}^{3 } - \frac{3}{2} {x}^{2} + 8)dx = \frac{1}{3} \times \frac{ {x}^{4} }{4} - \frac{3}{2} \times \frac{ {x}^{3} }{3} + 8x + C = \frac{ {x}^{4} }{12} - \frac{ {x}^{3} }{2} + 8x + C$

$2)∫(2 - 3 \sqrt{x} )^{2} dx = ∫(4 - 12 \sqrt{x} + x)dx = 4x - 12 \times \frac{ {x}^{ \frac{3}{2} } }{ \frac{3}{2} } + \frac{ {x}^{2} }{2} + C = 4x - 8x \sqrt{x} + \frac{ {x}^{2} }{2 } + C$

$3)∫( \frac{ {x}^{2} + 1 }{x} )dx = ∫( \frac{ {x}^{2} }{x} + \frac{1}{x} )dx = ∫(x + \frac{1}{x} )dx = \frac{ {x}^{2} }{2} + ln(x) + C$

$4)∫( \frac{ {x}^{ \frac{1}{2} } + {x}^{ \frac{1}{3} } }{ {x}^{ \frac{1}{4} } } )dx = ∫( {x}^{ \frac{1}{2} - \frac{1}{4} } + {x}^{ \frac{1}{3} - \frac{1}{4} } )dx = ∫( {x}^{ \frac{1}{4} } + {x}^{ \frac{1}{12} } )dx = \frac{ {x}^{ \frac{5}{4} } }{ \frac{5}{4} } + \frac{ {x}^{ \frac{13}{12} } }{ \frac{13}{12} } + C = \frac{4}{5} x \sqrt[4]{x} + \frac{12}{13} x \sqrt[12]{x} + C$

$5)∫(( {x}^{ \frac{1}{2} - \frac{1}{4} } - 3 {x}^{ \frac{2}{5} - \frac{1}{4} } + {x}^{ - \frac{1}{4} } )dx = ∫( {x}^{ \frac{1}{4} } - 3 {x}^{ \frac{3}{20} } + {x}^{ - \frac{1}{4} } )dx = \frac{4}{5} x \sqrt[4]{x} - 3 \frac{ {x}^{ \frac{23}{20} } }{ \frac{23}{20} } + \frac{ {x}^{ \frac{3}{4} } }{ \frac{3}{4} } + C = \frac{4}{5} x \sqrt[4]{x} - \frac{60}{23} x \sqrt[20]{ {x}^{3} } + \frac{4}{3} \sqrt[4]{ {x}^{3} } + C$

$6)∫ (\frac{3}{4} {x}^{2} - 2 \cos(x) )dx = \frac{3}{4} \times \frac{ {x}^{3} }{3} - 2 \sin(x) + C = \frac{ {x}^{3} }{4} - 2 \sin(x) + c$

$7)∫ \frac{ {( {x}^{2} + 1) }^{2} }{ {x}^{2} } dx = ∫ \frac{ {x}^{4} + 2 {x}^{2} + 1}{ {x}^{2} } dx = ∫( {x}^{2} + 2 + \frac{1}{ {x}^{2} } )dx = \frac{ {x}^{3} }{3} + 2x + \frac{ {x}^{ - 1} }{ - 1} + C = \frac{ {x}^{3} }{3} + 2x - \frac{1}{x} + C$

$8)∫ \frac{(1 + x) ^{3} }{ {x}^{2} } dx = ∫ \frac{ {x}^{3} + 3 {x}^{2} +3x + 1 }{ {x}^{2} } dx = ∫(x + 3 + \frac{3}{x} + {x}^{ - 2} )dx = \frac{ {x}^{2} }{2} + 3x + 3 ln(x) - \frac{1}{x} + C$

$9)∫(5 {e}^{x} + 4)dx = 5 {e}^{x} + 4x + C$

$10)∫( \frac{1}{x} + 2 {x}^{ - 2} + 3 {x}^{ - 3} dx = ln(x) - \frac{2}{x} + 3 \times \frac{ {x}^{ - 2} }{( - 2)} + C = ln(x) - \frac{2}{x} - \frac{3}{2 {x}^{2} } + C$