∫ [a^2-5a+6/a^2+7a+6:(2a+10/a+1-a-1)+1/a+3]*2a^2+5a-3/2a^2 eching

$$\int [ \frac { a ^ { 2 } - 5 a + 6 } { a ^ { 2 } + 7 a + 6 } : ( \frac { 2 a + 10 } { a + 1 } - a - 1 ) + \frac { 1 } { a + 3 } ] \cdot \frac { 2 a ^ { 2 } + 5 a - 3 } { 2 a ^ { 2 } }$$

Baholash

$$\frac{8ax-4x}{\left(a+6\right)a^{2}}+С$$

Yechim qadamlari

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. -a-1 ni \frac{a+1}{a+1} marotabaga ko'paytirish.

$$\int \left(\frac{\frac{a^{2}-5a+6}{a^{2}+7a+6}}{\frac{2a+10}{a+1}+\frac{\left(-a-1\right)\left(a+1\right)}{a+1}}+\frac{1}{a+3}\right)\times \frac{2a^{2}+5a-3}{2a^{2}}\mathrm{d}x$$

\frac{2a+10}{a+1} va \frac{\left(-a-1\right)\left(a+1\right)}{a+1} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\int \left(\frac{\frac{a^{2}-5a+6}{a^{2}+7a+6}}{\frac{2a+10+\left(-a-1\right)\left(a+1\right)}{a+1}}+\frac{1}{a+3}\right)\times \frac{2a^{2}+5a-3}{2a^{2}}\mathrm{d}x$$

2a+10+\left(-a-1\right)\left(a+1\right) ichidagi ko‘paytirishlarni bajaring.

$$\int \left(\frac{\frac{a^{2}-5a+6}{a^{2}+7a+6}}{\frac{2a+10-a^{2}-a-a-1}{a+1}}+\frac{1}{a+3}\right)\times \frac{2a^{2}+5a-3}{2a^{2}}\mathrm{d}x$$

2a+10-a^{2}-a-a-1 kabi iboralarga o‘xshab birlashtiring.

$$\int \left(\frac{\frac{a^{2}-5a+6}{a^{2}+7a+6}}{\frac{9-a^{2}}{a+1}}+\frac{1}{a+3}\right)\times \frac{2a^{2}+5a-3}{2a^{2}}\mathrm{d}x$$

\frac{a^{2}-5a+6}{a^{2}+7a+6} ni \frac{9-a^{2}}{a+1} ga bo'lish \frac{a^{2}-5a+6}{a^{2}+7a+6} ga k'paytirish \frac{9-a^{2}}{a+1} ga qaytarish.

$$\int \left(\frac{\left(a^{2}-5a+6\right)\left(a+1\right)}{\left(a^{2}+7a+6\right)\left(9-a^{2}\right)}+\frac{1}{a+3}\right)\times \frac{2a^{2}+5a-3}{2a^{2}}\mathrm{d}x$$

\frac{\left(a^{2}-5a+6\right)\left(a+1\right)}{\left(a^{2}+7a+6\right)\left(9-a^{2}\right)} ichida hali faktorlanmagan ifodalarni faktorlang.

$$\int \left(\frac{\left(a-3\right)\left(a-2\right)\left(a+1\right)}{\left(a-3\right)\left(-a-3\right)\left(a+1\right)\left(a+6\right)}+\frac{1}{a+3}\right)\times \frac{2a^{2}+5a-3}{2a^{2}}\mathrm{d}x$$

Surat va maxrajdagi ikkala \left(a-3\right)\left(a+1\right) ni qisqartiring.

$$\int \left(\frac{a-2}{\left(-a-3\right)\left(a+6\right)}+\frac{1}{a+3}\right)\times \frac{2a^{2}+5a-3}{2a^{2}}\mathrm{d}x$$

Ifodalarni qo‘shish yoki ayirish uchun ularni yoyib, maxrajlarini bir xil qiling. \left(-a-3\right)\left(a+6\right) va a+3 ning eng kichik umumiy karralisi \left(a+3\right)\left(a+6\right). \frac{a-2}{\left(-a-3\right)\left(a+6\right)} ni \frac{-1}{-1} marotabaga ko'paytirish. \frac{1}{a+3} ni \frac{a+6}{a+6} marotabaga ko'paytirish.

$$\int \left(\frac{-\left(a-2\right)}{\left(a+3\right)\left(a+6\right)}+\frac{a+6}{\left(a+3\right)\left(a+6\right)}\right)\times \frac{2a^{2}+5a-3}{2a^{2}}\mathrm{d}x$$

\frac{-\left(a-2\right)}{\left(a+3\right)\left(a+6\right)} va \frac{a+6}{\left(a+3\right)\left(a+6\right)} da bir xil maxraji bor, ularning suratini qo‘shish orqali qo‘shing.

$$\int \frac{-\left(a-2\right)+a+6}{\left(a+3\right)\left(a+6\right)}\times \frac{2a^{2}+5a-3}{2a^{2}}\mathrm{d}x$$

-\left(a-2\right)+a+6 ichidagi ko‘paytirishlarni bajaring.

$$\int \frac{-a+2+a+6}{\left(a+3\right)\left(a+6\right)}\times \frac{2a^{2}+5a-3}{2a^{2}}\mathrm{d}x$$

-a+2+a+6 kabi iboralarga o‘xshab birlashtiring.

$$\int \frac{8}{\left(a+3\right)\left(a+6\right)}\times \frac{2a^{2}+5a-3}{2a^{2}}\mathrm{d}x$$

Suratni maxrajga va maxrajini suratga ko‘paytirish orqali \frac{8}{\left(a+3\right)\left(a+6\right)} ni \frac{2a^{2}+5a-3}{2a^{2}} ga ko‘paytiring.

$$\int \frac{8\left(2a^{2}+5a-3\right)}{\left(a+3\right)\left(a+6\right)\times 2a^{2}}\mathrm{d}x$$

Surat va maxrajdagi ikkala 2 ni qisqartiring.

$$\int \frac{4\left(2a^{2}+5a-3\right)}{\left(a+3\right)\left(a+6\right)a^{2}}\mathrm{d}x$$

\frac{4\left(2a^{2}+5a-3\right)}{\left(a+3\right)\left(a+6\right)a^{2}} ichida hali faktorlanmagan ifodalarni faktorlang.

$$\int \frac{4\left(2a-1\right)\left(a+3\right)}{\left(a+3\right)\left(a+6\right)a^{2}}\mathrm{d}x$$

Surat va maxrajdagi ikkala a+3 ni qisqartiring.

$$\int \frac{4\left(2a-1\right)}{\left(a+6\right)a^{2}}\mathrm{d}x$$

4 ga 2a-1 ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\int \frac{8a-4}{\left(a+6\right)a^{2}}\mathrm{d}x$$

a+6 ga a^{2} ni ko'paytirish orqali distributiv xususiyatdan foydalanish.

$$\int \frac{8a-4}{a^{3}+6a^{2}}\mathrm{d}x$$

\int a\mathrm{d}x=ax umumiy integrallar qoidasi jadvalidan foydalanib, \frac{8a-4}{a^{3}+6a^{2}} integralini toping.

$$\frac{8a-4}{a^{3}+6a^{2}}x$$

Qisqartirish.

$$\frac{\left(8a-4\right)x}{a^{3}+6a^{2}}$$

Агар F\left(x\right)f\left(x\right) ning dastlabki holati boʻlsa, u holatda f\left(x\right) ning barcha dastlabki holatlari toʻplami F\left(x\right)+C tarafidan belgilanadi. Shu sababli natijaga C\in \mathrm{R} integrallash konstantasini qoʻshing.

$$\frac{\left(8a-4\right)x}{a^{3}+6a^{2}}+С$$

x ga nisbatan hosilani topish

$$\frac{4\left(2a-1\right)}{\left(a+6\right)a^{2}}$$