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شائع >

integral of (-1)/([(ax)^2-b^2]^{3/2)}

الحلّ

\int \frac{- 1}{[ ( a x ) ^{2} - b ^{2} ] ^{\frac{3}{2}}} d x

الحلّ

\frac{x}{b ^{2} ( a ^{2} x ^{2} - b ^{2} ) ^{\frac{1}{2}}} + C

خطوات الحلّ

\int \frac{- 1}{[ ( a x ) ^{2} - b ^{2} ] ^{\frac{3}{2}}} d x

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

:قم بإخراج الثابت

= - \int \frac{1}{[ ( a x ) ^{2} - b ^{2} ] ^{\frac{3}{2}}} d x

فعّل طريقة التكامل بالتعويض

: \int \frac{1}{a ( u ^{2} - b ^{2} ) ^{\frac{3}{2}}} d u

= - \int \frac{1}{a ( u ^{2} - b ^{2} ) ^{\frac{3}{2}}} d u

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

:قم بإخراج الثابت

= - \frac{1}{a} \cdot \int \frac{1}{( u ^{2} - b ^{2} ) ^{\frac{3}{2}}} d u

Apply Trigonometric Substitution

: \int \frac{sec ( v )}{b ^{2} tan ^{2} ( v )} d v

= - \frac{1}{a} \cdot \int \frac{sec ( v )}{b ^{2} tan ^{2} ( v )} d v

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

:قم بإخراج الثابت

= - \frac{1}{a} \cdot \frac{1}{b ^{2}} \cdot \int \frac{sec ( v )}{tan ^{2} ( v )} d v

Rewrite using trig identities

= - \frac{1}{a} \cdot \frac{1}{b ^{2}} \cdot \int \frac{cos ( v )}{sin ^{2} ( v )} d v

فعّل طريقة التكامل بالتعويض

: \int \frac{1}{w ^{2}} d w

= - \frac{1}{a} \cdot \frac{1}{b ^{2}} \cdot \int \frac{1}{w ^{2}} d w

استعمل قانون الأسس

: - \frac{1}{w}

= - \frac{1}{a} \cdot \frac{1}{b ^{2}} (- \frac{1}{w})

استبدل مجددًا

= - \frac{1}{a} \cdot \frac{1}{b ^{2}} (- \frac{1}{sin ( arcsec ( \frac{1}{b} a x ) )})

- \frac{1}{a} \cdot \frac{1}{b ^{2}} (- \frac{1}{sin ( arcsec ( \frac{1}{b} a x ) )})

بسّط:

\frac{x}{b ^{2} ( a ^{2} x ^{2} - b ^{2} ) ^{\frac{1}{2}}}

= \frac{x}{b ^{2} ( a ^{2} x ^{2} - b ^{2} ) ^{\frac{1}{2}}}

أضف ثابت التكامل للحل

= \frac{x}{b ^{2} ( a ^{2} x ^{2} - b ^{2} ) ^{\frac{1}{2}}} + C

رسم

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