Question

Найдите первообразную функции f(x) = 4x^5 - 5x для которой F(1) = -2

Answer 1

Находим первообразную:

$\displaystyle F(x) = \int( {4x}^{5} - 5x)dx = \not4 \int \bigg(\frac{ {x}^{6} }{ \not6} - \frac{5}{2} {x}^{2} \bigg)dx =$

$\displaystyle = \frac{2}{3} {x}^{6} - \frac{5}{2} {x}^{2} + C$

F(x) = -2; x = 1

$\displaystyle\frac{2}{3} \: * \: {1}^{6} - \frac{5}{2} \: * \: {1}^{2} + C = - 2$

$\displaystyle \frac{\stackrel{2/}{}2}{3} - \frac{\stackrel{3/}{}5}{2} + C = - 2$

$\displaystyle - \frac{11}{6} + C = - 2$

$\displaystyle C = \stackrel{3/}{} -2 + \frac{11}{6}$

$\displaystyle C = \frac{5}{6}$

$\displaystyle \boldsymbol F(x) = \frac{2}{3} {x}^{6} - \frac{5}{2} {x}^{2} + \frac{5}{6}$