Question

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Answer 1

ЛИСТ 1


1.

$\int{x} \, dx = \int{x^{[1]}} \, dx = \frac{1}{1+[1]} \cdot x^{[1]+1} + C = \frac{ x^2 }{2} + C \ ;$

$\int{ \frac{3}{2} \sin{x} } \, dx = \frac{3}{2} \int{ \sin{x} } \, dx = \frac{3}{2} \cdot ( - \cos{x} ) + C = -\frac{3}{2} \cos{x} + C \ ;$

$\int{ -\frac{1}{ \sin^2{x} } } \, dx = \int{ \frac{ d ( \pi/2 - x ) }{ \cos^2{ ( \pi/2 - x ) } } } = tg{ ( \frac{ \pi }{2} - x ) } + C = ctg{ x } + C \ ;$

$\int{ \cos{2x} } \, dx = \frac{1}{2} \int{ \cos{2x} } \, d2x = \frac{1}{2} \sin{2x} + C = \frac{ \sin{2x} }{2} + C \ ;$

$\int{ e^{ 5x+1 } } \, dx = \frac{1}{5} \int{ e^{ 5x+1 } } \, d(5x+1) = \frac{1}{5} e^{ 5x+1 } + C = \frac{ e^{ 5x+1 } }{5} + C \ ;$

$\int{ \frac{ 3dx }{ \cos^2{ ( 3x-1 ) } } } = \int{ \frac{ d(3x-1) }{ \cos^2{ ( 3x-1 ) } } } = tg{ ( 3x-1 ) } + C \ ;$



2.

$F(x) = \int{ f(x) } \, dx = \int{ ( \cos{2x} + \sin{3x} ) } \, dx = \\\\ = \int{ \cos{2x} } \, dx + \int{ \sin{3x} } \, dx = \frac{1}{2} \int{ \cos{2x} } \, d2x + \frac{1}{3} \int{ \sin{3x} } \, d3x = \\\\ = \frac{1}{2} \sin{2x} - \frac{1}{3} \cos{3x} + C = \frac{ \sin{2x} }{2} - \frac{ \cos{3x} }{3} + C \ ;$


$F(x) = \int{ f(x) } \, dx = \int{ \frac{2}{ (3x-1)^3 } } \, dx = \frac{2}{3} \int{ \frac{ d(3x-1) }{ (3x-1)^3 } } = \\\\ = \frac{2}{3} \int{ (3x-1)^{-3} } \, d(3x-1) = \frac{2}{3} \cdot \frac{1}{-3+1} \cdot (3x-1)^{-3+1} + C = \\\\ = \frac{2}{3} \cdot \frac{1}{-2} \cdot (3x-1)^{-2} + C = -\frac{1}{ 3 (3x-1)^2 } + C \ ;$





ЛИСТ 2


1.

$\int{ -\frac{3}{ \cos^2{x} } } \, dx = -3 \int{ \frac{dx}{ \cos^2{x} } } = -3 tg{x} + C \ ;$

$\int{ -\sin{3x} } \, dx = \frac{1}{3} \int{ -\sin{3x} } \, d3x = \frac{1}{3} \cos{3x} + C = \frac{ \cos{3x} }{3} + C \ ;$

$\int{ 4 \cos{ ( 4x + 2 ) } } \, dx = \int{ \cos{ ( 4x + 2 ) } } \, d( 4x + 2 ) = \sin{ ( 4x + 2 ) } + C \ ;$

$\int{ \frac{1}{ 7x-1 } } \, dx = \frac{1}{7} \int{ \frac{ d( 7x-1 ) }{ 7x-1 } } = \frac{1}{7} \ln{ | 7x-1 | } + C = \frac{ \ln{ | 7x-1 | } }{7} + C \ ;$


2.

$F(x) = \int{ f(x) } \, dx = \int{ ( \frac{1}{ \cos^2{2x} } - \frac{1}{ \sin^2{x} } ) } \, dx = \frac{1}{2} \int{ \frac{ d2x }{ \cos^2{2x} } } + \int{ \frac{ d( \pi/2 - x ) }{ \cos^2{ ( \pi/2 - x } ) } } = \\\\ = \frac{1}{2} tg{2x} + tg{ ( \frac{ \pi }{2} - x ) } + C = \frac{ tg{2x} }{2} + ctg{x} + C \ ;$


$F(x) = \int{ f(x) } \, dx = \int{ 2 \sqrt[3]{ 3 - 2x } } \, dx = - \int{ ( 3 - 2x )^{1/3} } \, d( 3 - 2x ) = \\\\ = - \frac{1}{ 1/3 + 1 } \cdot ( 3 - 2x )^{ 1/3 + 1 } + C = - \frac{1}{ 4/3 } \cdot ( 3 - 2x )^{ 4/3 } + C = - \frac{3}{4} \sqrt[3]{ ( 3 - 2x )^4 } + C \ ;$