evaluate the integral. \n∫₀¹ (7 - 8v³ + 16v⁷) dv

Answer:

4

Explanation:

Step1: Aplicar reglas de integración

$$\int_{0}^{1}(7 - 8v^{3}+16v^{7})dv=\int_{0}^{1}7dv-\int_{0}^{1}8v^{3}dv+\int_{0}^{1}16v^{7}dv$$

Step2: Integrar cada término

• $\int_{0}^{1}7dv = 7v\big|_{0}^{1}=7(1 - 0)=7$
• $\int_{0}^{1}8v^{3}dv=8\times\frac{v^{4}}{4}\big|{0}^{1}=2v^{4}\big|{0}^{1}=2(1 - 0)=2$
• $\int_{0}^{1}16v^{7}dv=16\times\frac{v^{8}}{8}\big|{0}^{1}=2v^{8}\big|{0}^{1}=2(1 - 0)=2$

Step3: Calcular el resultado final

$7-2 + 2=4$