What is the integral of (x^2)/((1-x^2)^{5/2)} ?

The integral of (x^2)/((1-x^2)^{5/2)} is (x^3)/(3(1-x^2)^{3/2)}+C

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integral of (x^2)/((1-x^2)^{5/2)}

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

\frac{x ^{3}}{3 ( 1 - x ^{2} ) ^{\frac{3}{2}}} + C

Solution steps

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

Apply Trigonometric Substitution

= \int \frac{sin ^{2} ( u )}{cos ^{4} ( u )} d u

Rewrite using trig identities

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

Apply u-substitution

= \int v^{2} d v

Apply the Power Rule

= \frac{v ^{3}}{3}

Substitute back

= \frac{tan ^{3} ( arcsin ( x ) )}{3}

Simplify

\frac{tan ^{3} ( arcsin ( x ) )}{3} : \frac{x ^{3}}{3 ( 1 - x ^{2} ) ^{\frac{3}{2}}}

= \frac{x ^{3}}{3 ( 1 - x ^{2} ) ^{\frac{3}{2}}}

Add a constant to the solution

= \frac{x ^{3}}{3 ( 1 - x ^{2} ) ^{\frac{3}{2}}} + C

\frac{e ^{- 2 s}}{s ^{2}}

x \to - 2 lim (8)

f (x) = \frac{168}{( x + 4 ) ^{5}}

f (x) = x e^{x}

\int ln (7 x^{5}) d x

What is the integral of (x^2)/((1-x^2)^{5/2)} ?
The integral of (x^2)/((1-x^2)^{5/2)} is (x^3)/(3(1-x^2)^{3/2)}+C