Lista - Integrais Impróprias - Cálculo 2 2021 2

Integrais Impr´oprias FAMAT-UFU P´agina 1 Este trabalho deve ser resolvido e entregue dia 05/07/22 INTEGRAIS IMPR´OPRIAS Determine se cada integral ´e convergente ou divergente. Calcule aquelas que s˜ao convergentes. a) ∞∫ 3 1 (x − 2) 3 2 dx b) 0∫ −∞ 1 3 − 4xdx c) ∞∫ −∞ xe−x2dx d) ∞∫ e 1 x (ln x)3dx e) ∞∫ 1 ln x x dx f) 1∫ 0 3 x5dx g) 0∫ −∞ xe2xdx h) 14∫ −2 1 4√x + 2dx i) 0∫ −∞ 1 1 + x2dx j) +∞ ∫ 2 1 x2 − 1dx k) 2∫ 0 1 √2 − xdx l) 1∫ 0 x √ 1 − x2dx m) π 2∫ − π 2 tg (x) dx n) 1∫ −1 1 |x|dx lais@ufu.br sites.google.com/site/laisufu La´ıs Rodrigues e) \int_0^1 \frac{x}{\sqrt{1-x^2}} \, dx \ \text{substituição:} \ u = 1 - x^2 \\ xu dx = -\frac{du}{2} \, \ -du = -2x \, dx \\ -\int \frac{du}{2\cdot\sqrt{u}} = -\frac{1}{2} \int \frac{du}{\sqrt{u}} = -\frac{1}{2} (2\sqrt{u}) \\ dado \ que \ u = 1-x^2: \\ -\left|_{0}^{1} (x\cdot\sqrt{1-x^2}) \right. = -\sqrt{1-x^2}\right|_{0}^{1} = \sqrt{0} -(-\sqrt{1}) = 1 \\ m) \int_{-\pi/2}^{\pi/2} \text{tg}(x)\, dx \ \rightarrow \ função \ impar: \\ f(x) = \text{tg}(x) \\ f(-x) = \text{tg}(-x) = -\text{tg}(x) \\ logo : \int_{-\pi/2}^{\pi/2} \text{tg}(x)\, dx = \boxed{0} \\ n) \int_{-1}^{1} \frac{1}{|x|} \, dx \ \Rightarrow \int_{-1}^{0}\frac{1}{-x}\, dx + \int_{0}^{1}\frac{1}{x}\, dx \\ \text{(DIVERGE)} \ \Rightarrow \frac{1}{0} \\ \int_{-1}^{1} \frac{1}{|x|} \, dx \ \rightarrow \text{DIVERGE} \\