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شائع حساب التفاضل والتكامل >

integral of 3/((x^2+2)^2)

الحلّ

\int \frac{3}{( x ^{2} + 2 ) ^{2}} d x

الحلّ

\frac{3}{4 2} (arctan (\frac{x}{2}) + \frac{1}{2} sin (2 arctan (\frac{x}{2}))) + C

خطوات الحلّ

\int \frac{3}{( x ^{2} + 2 ) ^{2}} d x

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

:قم بإخراج الثابت

= 3 \cdot \int \frac{1}{( x ^{2} + 2 ) ^{2}} d x

فعّل طريقة التكامل بالتعويض

= 3 \cdot \int \frac{2}{( 2 u ^{2} + 2 ) ^{2}} d u

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

:قم بإخراج الثابت

= 32 \cdot \int \frac{1}{( 2 u ^{2} + 2 ) ^{2}} d u

\frac{1}{( 2 u ^{2} + 2 ) ^{2}}

حلل إلى عوامل

= 32 \cdot \int \frac{1}{4 ( u ^{2} + 1 ) ^{2}} d u

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

:قم بإخراج الثابت

= 32 \frac{1}{4} \cdot \int \frac{1}{( u ^{2} + 1 ) ^{2}} d u

Apply Trigonometric Substitution

= 32 \frac{1}{4} \cdot \int \frac{1}{sec ^{4} ( v ) cos ^{2} ( v )} d v

Rewrite using trig identities

= 32 \frac{1}{4} \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

= 32 \frac{1}{4} \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

= 32 \frac{1}{4} \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)

= 32 \frac{1}{4} \cdot \frac{1}{2} (v + \frac{1}{2} sin (2 v))

استبدل مجددًا

= 32 \frac{1}{4} \cdot \frac{1}{2} (arctan (\frac{x}{2}) + \frac{1}{2} sin (2 arctan (\frac{x}{2})))

32 \frac{1}{4} \cdot \frac{1}{2} (arctan (\frac{x}{2}) + \frac{1}{2} sin (2 arctan (\frac{x}{2})))

بسّط:

\frac{3}{4 2} (arctan (\frac{x}{2}) + \frac{1}{2} sin (2 arctan (\frac{x}{2})))

= \frac{3}{4 2} (arctan (\frac{x}{2}) + \frac{1}{2} sin (2 arctan (\frac{x}{2})))

أضف ثابت التكامل للحل

= \frac{3}{4 2} (arctan (\frac{x}{2}) + \frac{1}{2} sin (2 arctan (\frac{x}{2}))) + C

رسم

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