What is the integral of (x^2)/((9-x^2)^{3/2)} ?

The integral of (x^2)/((9-x^2)^{3/2)} is -arcsin(1/3 x)+x/((9-x^2)^{1/2)}+C

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integral of (x^2)/((9-x^2)^{3/2)}

\int \frac{x ^{2}}{( 9 - x ^{2} ) ^{\frac{3}{2}}} d x

- arcsin (\frac{1}{3} x) + \frac{x}{( 9 - x ^{2} ) ^{\frac{1}{2}}} + C

Solution steps

\int \frac{x ^{2}}{( 9 - x ^{2} ) ^{\frac{3}{2}}} d x

Apply Trigonometric Substitution

: \int tan^{2} (u) d u

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

Rewrite using trig identities

= \int - 1 + sec^{2} (u) d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= - \int 1 d u + \int sec^{2} (u) d u

\int 1 d u = u

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

= - u + tan (u)

Substitute back

u = arcsin (\frac{1}{3} x)

= - arcsin (\frac{1}{3} x) + tan (arcsin (\frac{1}{3} x))

Simplify

- arcsin (\frac{1}{3} x) + tan (arcsin (\frac{1}{3} x)) : - arcsin (\frac{1}{3} x) + \frac{x}{( 9 - x ^{2} ) ^{\frac{1}{2}}}

= - arcsin (\frac{1}{3} x) + \frac{x}{( 9 - x ^{2} ) ^{\frac{1}{2}}}

Add a constant to the solution

= - arcsin (\frac{1}{3} x) + \frac{x}{( 9 - x ^{2} ) ^{\frac{1}{2}}} + C

\int_{1}^{1000} \frac{37}{x ^{3}} d x

(\frac{1}{y ^{2}} - \frac{3}{y ^{4}}) (y + 7 y^{3})

\frac{\partial}{\partial x} (2 x e^{2 x y})

x \to + 7 lim (\frac{x}{x + 7})

\frac{\partial}{\partial x} (\frac{x e ^{y}}{3 z ^{2} + 1})

What is the integral of (x^2)/((9-x^2)^{3/2)} ?
The integral of (x^2)/((9-x^2)^{3/2)} is -arcsin(1/3 x)+x/((9-x^2)^{1/2)}+C