Question

Дам за ришения 25 бал
лов

Answer 1

Решение:

1.  ∫7 dx = 7x + C

$2.~~\int {x^8} \, dx = \dfrac{x^9}{9} + C$

$3.~~\int {\dfrac{1}{x}} \, dx = ln |x|+ C$

$4.~~\int {sinx} \, dx = -cosx+ C$

$5.~~\int {8e^x} \, dx = 8e^x+ C$

$6.~~\int {4cosx} \, dx = 4sinx+ C$

$7.~~\int {(7x-8)^4} \, dx = \dfrac{1}{7} \int {(7x-8)^4} \, d(7x-8)= \dfrac{(7x-8)^5}{35}+C$

$8.~~\int {(7x^2-3x^3+4x^5)} \, dx = \dfrac{7x^3}{3} - \dfrac{3x^4}{4}+\dfrac{2x^6}{3}+C$

$9.~~\int {sin(7x-\dfrac{\pi}{4}) } \, dx =\dfrac{1}{7} \int {sin(7x-\dfrac{\pi}{4}) } \, d(7x-\dfrac{\pi}{4})= -\dfrac{cos(7x-\dfrac{\pi}{4}) }{7} +C$

$10.~~\int {x\cdot 2^x } \, dx =\dfrac{2^x(x\cdot ln2 - 1)}{ln^22} +C$

Процесс решения 10:

Интегрируем по частям

∫f · g' dx = f · g - ∫f' · g dx

Здесь

f = x     g' = 2ˣ

f' = 1     $g = \dfrac{2^x}{ln2}$

$~\int {x\cdot 2^x } \, dx =\dfrac{x\cdot 2^x}{ln2} - \int {\dfrac{2^x}{ln2}} \, dx=\dfrac{x\cdot 2^x}{ln2} - \dfrac{ 2^x}{ln^22} = \dfrac{2^x(x\cdot ln2 - 1)}{ln^22} +C$