Question

пожалуйста помогите,срочно!!

Answer 1

$\text{a}) ~ \displaystyle \int (1-x)(x+1)^{4} \, dx = \left | {{u = 1 - x; ~ du = -dx} \atop {dv=(x+1)^{4}\, dx; ~ v = \dfrac{1}{5}(x+1)^{5} }} \right | =\\\\= \dfrac{1}{5} (1-x)(x+1)^{5} - \int \dfrac{1}{5}(x+1)^{5} }} \cdot(-dx) = \\\\=\dfrac{1}{5} (1-x)(x+1)^{5} + \frac{1}{5} \int (x+1)^{5}\,dx = \\\\=\dfrac{1}{5} (1-x)(x+1)^{5} + \frac{1}{5} \cdot \frac{(x+1)^{6}}{6} + C = \\\\=\dfrac{1}{5} (1-x)(x+1)^{5} + \frac{1}{30}(x+1)^{6} + C.$

$\text{b}) ~ \displaystyle S=\int\limits_{-1}^{1} (1-x)(x+1)^{4} \, dx = \left(\dfrac{1}{5} (1-x)(x+1)^{5} + \frac{1}{30}(x+1)^{6} \right) \Bigg|^{1}_{-1} =\\\\= \dfrac{1}{5} (1-1)(1+1)^{5} + \frac{1}{30}(1+1)^{6} - \left(\dfrac{1}{5} (1+1)(-1+1)^{5} + \frac{1}{30}(-1+1)^{6} \right)=\\\\= \dfrac{1}{30} \cdot 2^{6} - 0 = \frac{32}{15}$

$\text{c}) ~ V = \displaystyle \pi\int\limits^{1}_{-1}\left((1-x)(x+1)^{4}\right)^{2}\, dx = \pi\int\limits^{1}_{-1}(1-x)^{2}(x+1)^{8}\, dx$

$\displaystyle \int(1-x)^{2}(x+1)^{8}\, dx = \left | {{u=(1-x)^{2}; ~ du = (2x-2)dx} \atop {dv = (x+1)^{8}\, dx; ~ v = \dfrac{1}{9}(x+1)^{9} }} \right| =\\\\= \frac{1}{9} (1-x)^{2}(x+1)^{9} - \int \dfrac{1}{9}(x+1)^{9} } (2x-2)\,dx =\\\\= \left | {{w=2x-2; ~ dw = 2\,dx} \atop {dj = (x+1)^{9}\, dx; ~ j = \dfrac{1}{10}(x+1)^{10} }} \right| = \frac{1}{9} (1-x)^{2}(x+1)^{9} - \\\\- \frac{1}{9} \left(\frac{1}{10}(2x-2)(x+1)^{10} - \int \frac{1}{10}(x+1)^{10}\cdot 2 \, dx\right) =$

$= \dfrac{1}{9} (1-x)^{2}(x+1)^{9} - \dfrac{1}{45}(x-1)(x+1)^{10} + \dfrac{1}{495}(x+1)^{11} + C$

$V = \displaystyle \pi\left(\dfrac{1}{9} (1-x)^{2}(x+1)^{9} - \dfrac{1}{45}(x-1)(x+1)^{10} + \dfrac{1}{495}(x+1)^{11} \right) \Bigg|^{1}_{-1} =\\\\= \pi \left(\dfrac{1}{9} (1-1)^{2}(1+1)^{9} - \dfrac{1}{45}(1-1)(1+1)^{10} + \dfrac{1}{495}(1+1)^{11} -$

$- \left(\dfrac{1}{9} (1+1)^{2}(-1+1)^{9} - \dfrac{1}{45}(-1-1)(-1+1)^{10} + \dfrac{1}{495}(-1+1)^{11} \right) \Bigg) =$

$= \pi \left(\dfrac{1}{495} \cdot 2^{11} - 0\right) = \dfrac{2048 \pi}{495}$

Ответ:

• $\text{a}) ~ \displaystyle \dfrac{1}{5} (1-x)(x+1)^{5} + \frac{1}{30}(x+1)^{6} + C$

• $\text{b}) ~ \dfrac{32}{15}$ кв. кд.
• $\text{c}) ~ \dfrac{2048\pi}{495}$ куб. ед.

Использованные формулы:

$\displaystyle \int u\,dv = uv - \int v \, du$

$\displaystyle \int x^{a} \, dx = \frac{x^{a+1}}{a+1} + C, ~ a \neq -1$

$\displaystyle S = \int\limits_{a}^{b} f(x) \, dx = F(x) \Big|^{b}_{a} = F(b) - F(a)$

$V = \displaystyle \int\limits_{a}^{b}S(x) \, dx = \pi \int\limits_{a}^{b} f^{2}(x) \, dx$