∫ (from 0 to x) of (sint-t)dt eching

$$\int _ { 0 } ^ { x } ( \sin t - t ) d t$$

Baholash

$$\frac{-2\cos(x)-x^{2}+2}{2}$$

Yechim qadamlari

Avval noaniq integralni baholang.

$$\int \sin(t)-t\mathrm{d}t$$

Summani muddatma-muddat integratsiya qiling.

$$\int \sin(t)\mathrm{d}t+\int -t\mathrm{d}t$$

Har bir shartda konstantani qavsdan tashqariga oling.

$$\int \sin(t)\mathrm{d}t-\int t\mathrm{d}t$$

Natijani olish uchun umumiy integrallar jadvalidagi \int \sin(t)\mathrm{d}t=-\cos(t) integralidan foydalaning.

$$-\cos(t)-\int t\mathrm{d}t$$

k\neq -1 uchun integral \int t^{k}\mathrm{d}t=\frac{t^{k+1}}{k+1} boʻlgani uchun, \int t\mathrm{d}t integralni \frac{t^{2}}{2} bilan almashtiring. -1 ni \frac{t^{2}}{2} marotabaga ko'paytirish.

$$-\cos(t)-\frac{t^{2}}{2}$$

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

$$-\cos(x)-\frac{x^{2}}{2}-\left(-\cos(0)-\frac{0^{2}}{2}\right)$$

Qisqartirish.

$$-\cos(x)-\frac{x^{2}}{2}+1$$

x ga nisbatan hosilani topish

$$\sin(x)-x$$