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

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

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

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

\frac{5 x ^{3}}{3 ( 1 - x ^{2} ) ^{\frac{3}{2}}} + C

Solution steps

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

Take the constant out:

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

= 5 \cdot \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

= 5 \cdot \int \frac{sin ^{2} ( u )}{cos ^{4} ( u )} d u

Rewrite using trig identities

= 5 \cdot \int \frac{tan ^{2} ( u )}{cos ^{2} ( u )} d u

Apply u-substitution

: \int v^{2} d v

= 5 \cdot \int v^{2} d v

Apply the Power Rule

: \frac{v ^{3}}{3}

= 5 \cdot \frac{v ^{3}}{3}

Substitute back

= 5 \cdot \frac{tan ^{3} ( arcsin ( x ) )}{3}

Simplify

5 \cdot \frac{tan ^{3} ( arcsin ( x ) )}{3} : \frac{5 x ^{3}}{3 ( 1 - x ^{2} ) ^{\frac{3}{2}}}

= \frac{5 x ^{3}}{3 ( 1 - x ^{2} ) ^{\frac{3}{2}}}

Add a constant to the solution

= \frac{5 x ^{3}}{3 ( 1 - x ^{2} ) ^{\frac{3}{2}}} + C

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