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شائع حساب التفاضل والتكامل >

integral of sqrt(x)sqrt(9-x)

الحلّ

\int x 9 - x d x

الحلّ

\frac{81}{16} (2 arcsin (\frac{2}{9} x - 1) + sin (2 arcsin (\frac{2}{9} x - 1))) + C

خطوات الحلّ

\int x 9 - x d x

a^{n} \cdot b^{n} = (a \cdot b)^{n}

:فعّل قانون القوى

x 9 - x = x (9 - x)

= \int x (9 - x) d x

x (9 - x)

وسٌع:

9 x - x^{2}

= \int 9 x - x^{2} d x

9 x - x^{2}

إكمال لمربّع:

- (x - \frac{9}{2})^{2} + \frac{81}{4}

= \int - (x - \frac{9}{2})^{2} + \frac{81}{4} d x

فعّل طريقة التكامل بالتعويض

= \int \frac{- 4 u ^{2} + 81}{2} d u

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

:قم بإخراج الثابت

= \frac{1}{2} \cdot \int - 4 u^{2} + 81 d u

Apply Trigonometric Substitution

= \frac{1}{2} \cdot \int \frac{81}{2} cos^{2} (v) d v

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

:قم بإخراج الثابت

= \frac{1}{2} \cdot \frac{81}{2} \cdot \int cos^{2} (v) d v

Rewrite using trig identities

= \frac{1}{2} \cdot \frac{81}{2} \cdot \int \frac{1 + cos ( 2 v )}{2} d v

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

:قم بإخراج الثابت

= \frac{1}{2} \cdot \frac{81}{2} \cdot \frac{1}{2} \cdot \int 1 + cos (2 v) d v

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

:فعّل الميزة الخطيّة

= \frac{1}{2} \cdot \frac{81}{2} \cdot \frac{1}{2} (\int 1 d v + \int cos (2 v) d v)

\int 1 d v = v

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

= \frac{1}{2} \cdot \frac{81}{2} \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

استبدل مجددًا

= \frac{1}{2} \cdot \frac{81}{2} \cdot \frac{1}{2} (arcsin (\frac{2}{9} (x - \frac{9}{2})) + \frac{1}{2} sin (2 arcsin (\frac{2}{9} (x - \frac{9}{2}))))

\frac{1}{2} \cdot \frac{81}{2} \cdot \frac{1}{2} (arcsin (\frac{2}{9} (x - \frac{9}{2})) + \frac{1}{2} sin (2 arcsin (\frac{2}{9} (x - \frac{9}{2}))))

بسّط:

\frac{81}{16} (2 arcsin (\frac{2}{9} x - 1) + sin (2 arcsin (\frac{2}{9} x - 1)))

= \frac{81}{16} (2 arcsin (\frac{2}{9} x - 1) + sin (2 arcsin (\frac{2}{9} x - 1)))

أضف ثابت التكامل للحل

= \frac{81}{16} (2 arcsin (\frac{2}{9} x - 1) + sin (2 arcsin (\frac{2}{9} x - 1))) + C

رسم

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