∫ (from 1 to 4) of left(-x+4/xright) wrt x eching

$$\int_{ 1 }^{ 4 } \left( -x+ \frac{ 4 }{ x } \right) d x$$

Baholash

$$8\ln(2)-\frac{15}{2}\approx -1,954822556$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int -x+\frac{4}{x}\mathrm{d}x$$

Summani muddatma-muddat integratsiya qiling.

$$\int -x\mathrm{d}x+\int \frac{4}{x}\mathrm{d}x$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$-\int x\mathrm{d}x+4\int \frac{1}{x}\mathrm{d}x$$

k\neq -1 uchun integral \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} boʻlgani uchun, \int x\mathrm{d}x integralni \frac{x^{2}}{2} bilan almashtiring. -1 ni \frac{x^{2}}{2} marotabaga ko'paytirish.

$$-\frac{x^{2}}{2}+4\int \frac{1}{x}\mathrm{d}x$$

Natijani olish uchun umumiy integrallar jadvalidagi \int \frac{1}{x}\mathrm{d}x=\ln(|x|) integralidan foydalaning.

$$-\frac{x^{2}}{2}+4\ln(|x|)$$

Xos integral bu integral hisoblashning yuqori chegarasida hisoblangan ifodaning boshlangʻich holatidan chiqarib tashlagan holda integral hisoblashning quyi chegarasida hisoblangan ifodaning boshlangʻich holatidir.

$$-\frac{4^{2}}{2}+4\ln(|4|)-\left(-\frac{1^{2}}{2}+4\ln(|1|)\right)$$

Qisqartirish.

$$-\frac{15}{2}+8\ln(2)$$