integral of (p(x))/((x-2)(x^2+9))

الحلّ

\int \frac{p ( x )}{( x - 2 ) ( x ^{2} + 9 )} d x

p (\frac{2}{13} ln ∣ x - 2 ∣ + \frac{1}{13} (- ln x^{2} + 9 + 3 arctan (\frac{x}{3}))) + C

خطوات الحلّ

\int \frac{p x}{( x - 2 ) ( x ^{2} + 9 )} d x

\int a \cdot f (x) d x = a \cdot \int f (x) d x

:قم بإخراج الثابت

= p \cdot \int \frac{x}{( - 2 + x ) ( 9 + x ^{2} )} d x

\frac{x}{( - 2 + x ) ( 9 + x ^{2} )}

خذ الكسر الجزئي لـ:

\frac{2}{13 ( x - 2 )} + \frac{- 2 x + 9}{13 ( x ^{2} + 9 )}

= p \cdot \int \frac{2}{13 ( x - 2 )} + \frac{- 2 x + 9}{13 ( x ^{2} + 9 )} d x

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

:فعّل الميزة الخطيّة

= p (\int \frac{2}{13 ( x - 2 )} d x + \int \frac{- 2 x + 9}{13 ( x ^{2} + 9 )} d x)

\int \frac{2}{13 ( x - 2 )} d x = \frac{2}{13} ln ∣ x - 2 ∣

\int \frac{- 2 x + 9}{13 ( x ^{2} + 9 )} d x = \frac{1}{13} (- ln x^{2} + 9 + 3 arctan (\frac{x}{3}))

= p (\frac{2}{13} ln ∣ x - 2 ∣ + \frac{1}{13} (- ln x^{2} + 9 + 3 arctan (\frac{x}{3})))

أضف ثابت التكامل للحل

= p (\frac{2}{13} ln ∣ x - 2 ∣ + \frac{1}{13} (- ln x^{2} + 9 + 3 arctan (\frac{x}{3}))) + C

\int 50 - 8 x - 26 d x

\int \frac{( x ^{2} - 1 )}{x} d x

\int x^{2} (x^{2} + 6)^{\frac{- 1}{2}} d x

\int - x^{2} + 7 d x

\int \frac{1}{1 + a x} d x