What is the integral of (x^2)/((25-x^2)^{5/2)} ?

The integral of (x^2)/((25-x^2)^{5/2)} is (x^3)/(75(25-x^2)^{3/2)}+C

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integral of (x^2)/((25-x^2)^{5/2)}

\int \frac{x ^{2}}{( 25 - x ^{2} ) ^{\frac{5}{2}}} d x

\frac{x ^{3}}{75 ( 25 - x ^{2} ) ^{\frac{3}{2}}} + C

Solution steps

\int \frac{x ^{2}}{( 25 - x ^{2} ) ^{\frac{5}{2}}} d x

Multiply by the conjugate of

25 - x^{2} : \frac{625 - x ^{4}}{25 + x ^{2}}

= \int \frac{x ^{2}}{( \frac{625 - x ^{4}}{25 + x ^{2}} ) ^{\frac{5}{2}}} d x

Apply Integral Substitution

= \int \frac{u ^{2}}{25 ( - ( u + 1 ) ( u - 1 ) ) ^{\frac{5}{2}}} d u

Take the constant out:

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

= \frac{1}{25} \cdot \int \frac{u ^{2}}{( - ( u + 1 ) ( u - 1 ) ) ^{\frac{5}{2}}} d u

Multiply by the conjugate of

u + 1 : \frac{u ^{2} - 1}{u - 1}

= \frac{1}{25} \cdot \int \frac{u ^{2}}{( - \frac{u ^{2} - 1}{u - 1} ( u - 1 ) ) ^{\frac{5}{2}}} d u

Multiply

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

= \frac{1}{25} \cdot \int \frac{u ^{2}}{( - u ^{2} + 1 ) ^{\frac{5}{2}}} d u

Apply Trigonometric Substitution

= \frac{1}{25} \cdot \int \frac{sin ^{2} ( v )}{cos ^{4} ( v )} d v

Rewrite using trig identities

= \frac{1}{25} \cdot \int \frac{tan ^{2} ( v )}{cos ^{2} ( v )} d v

Apply u-substitution

= \frac{1}{25} \cdot \int w^{2} d w

Apply the Power Rule

= \frac{1}{25} \cdot \frac{w ^{3}}{3}

Substitute back

= \frac{1}{25} \cdot \frac{tan ^{3} ( arcsin ( \frac{x}{5} ) )}{3}

Simplify

\frac{1}{25} \cdot \frac{tan ^{3} ( arcsin ( \frac{x}{5} ) )}{3} : \frac{x ^{3}}{75 ( 25 - x ^{2} ) ^{\frac{3}{2}}}

= \frac{x ^{3}}{75 ( 25 - x ^{2} ) ^{\frac{3}{2}}}

Add a constant to the solution

= \frac{x ^{3}}{75 ( 25 - x ^{2} ) ^{\frac{3}{2}}} + C

What is the integral of (x^2)/((25-x^2)^{5/2)} ?
The integral of (x^2)/((25-x^2)^{5/2)} is (x^3)/(75(25-x^2)^{3/2)}+C