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شائع >

integral of (x^2)/(sqrt(a-x^2))

الحلّ

\int \frac{x ^{2}}{a - x ^{2}} d x

الحلّ

\frac{1}{2} a (arcsin (\frac{x}{a}) - \frac{1}{2} sin (2 arcsin (\frac{x}{a}))) + C

خطوات الحلّ

\int \frac{x ^{2}}{a - x ^{2}} d x

Apply Trigonometric Substitution

: \int a sin^{2} (u) d u

= \int a sin^{2} (u) d u

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

:قم بإخراج الثابت

= a \cdot \int sin^{2} (u) d u

Rewrite using trig identities

= a \cdot \int \frac{1 - cos ( 2 u )}{2} d u

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

:قم بإخراج الثابت

= a \frac{1}{2} \cdot \int 1 - cos (2 u) d u

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

:فعّل الميزة الخطيّة

= a \frac{1}{2} (\int 1 d u - \int cos (2 u) d u)

\int 1 d u = u

\int cos (2 u) d u = \frac{1}{2} sin (2 u)

= a \frac{1}{2} (u - \frac{1}{2} sin (2 u))

u = arcsin (\frac{1}{a} x)

استبدل مجددًا

= a \frac{1}{2} (arcsin (\frac{1}{a} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{a} x)))

a \frac{1}{2} (arcsin (\frac{1}{a} x) - \frac{1}{2} sin (2 arcsin (\frac{1}{a} x)))

بسّط:

\frac{1}{2} a (arcsin (\frac{x}{a}) - \frac{1}{2} sin (2 arcsin (\frac{x}{a})))

= \frac{1}{2} a (arcsin (\frac{x}{a}) - \frac{1}{2} sin (2 arcsin (\frac{x}{a})))

أضف ثابت التكامل للحل

= \frac{1}{2} a (arcsin (\frac{x}{a}) - \frac{1}{2} sin (2 arcsin (\frac{x}{a}))) + C

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