integral of (x^4)/((4-x^2)^{7/2)}

الحلّ

\int \frac{x ^{4}}{( 4 - x ^{2} ) ^{\frac{7}{2}}} d x

\frac{x ^{5}}{20 ( 4 - x ^{2} ) ^{\frac{5}{2}}} + C

خطوات الحلّ

\int \frac{x ^{4}}{( 4 - x ^{2} ) ^{\frac{7}{2}}} d x

Apply Trigonometric Substitution

= \int \frac{sin ^{4} ( u )}{4 cos ^{6} ( u )} d u

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

:قم بإخراج الثابت

= \frac{1}{4} \cdot \int \frac{sin ^{4} ( u )}{cos ^{6} ( u )} d u

Rewrite using trig identities

= \frac{1}{4} \cdot \int \frac{tan ^{4} ( u )}{cos ^{2} ( u )} d u

\frac{1}{cos ( x )} = sec (x)

:Use the basic trigonometric identity

= \frac{1}{4} \cdot \int tan^{4} (u) sec^{2} (u) d u

Rewrite using trig identities

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

فعّل طريقة التكامل بالتعويض

= \frac{1}{4} \cdot \int v^{4} d v

استعمل قانون الأسس

= \frac{1}{4} \cdot \frac{v ^{5}}{5}

استبدل مجددًا

= \frac{1}{4} \cdot \frac{tan ^{5} ( arcsin ( \frac{1}{2} x ) )}{5}

\frac{1}{4} \cdot \frac{tan ^{5} ( arcsin ( \frac{1}{2} x ) )}{5}

بسّط:

\frac{x ^{5}}{20 ( 4 - x ^{2} ) ^{\frac{5}{2}}}

= \frac{x ^{5}}{20 ( 4 - x ^{2} ) ^{\frac{5}{2}}}

أضف ثابت التكامل للحل

= \frac{x ^{5}}{20 ( 4 - x ^{2} ) ^{\frac{5}{2}}} + C

\int \frac{( 4 x - 5 )}{( x ^{2} - 2 x + 2 ) ^{\frac{3}{2}}} d x

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\int \frac{1}{u ( u + 1 )} d u

\int \frac{1}{3 x ^{2} - 5} d x